Solved: In a famous 18th-century problem, known as Buffons

Find the length of the arc of the semicubical parabola between the points and . (See Figure 5.)

Find the length of the arc of the parabola from 0, 0 1, 1 V

a) Set up an integral for the length of the arc of the hyperbola from the point to the point . (b) Use Simpsons Rule with to estimate the arc length.

Find the arc length function for the curve taking as the starting point.

Use the arc length formula (3) to find the length of the curve  , . Check your answer by noting that the curve is a line segment and calculating its length by the distance formula.

Use the arc length formula to find the length of the curve , . Check your answer by noting that the curve is part of a circle

Set up, but do not evaluate, an integral for the length of the curvey cos x 0 x 2

Set up, but do not evaluate, an integral for the length of the curvey xe 0 x 1

Set up, but do not evaluate, an integral for the length of the curvex y y 1 y 4 3

Set up, but do not evaluate, an integral for the length of the curvex 2 a2 y 2 b2 1

Find the length of the curve.y 1 6x 0 x 1 3

Find the length of the curve.0 x 2 y 0

Find the length of the curve.2 x 5 6 1

Find the length of the curve.2 y 4 8 1 4y

Find the length of the curve.x 1 y 9 1 11. 3 sy y 3 x

Find the length of the curve.y lncos x0 x  3

Find the length of the curve.y lnsec x0 x  4

Find the length of the curve.y 3 , 0 x 1 1

Find the length of the curve.0 x 1 y

Find the length of the curve.y sx x 2 sin1 (sx ) 0

Find the length of the curve.y e 0 x 1

Find the length of the curve.y ln a x b a 0 e x 1 ex 1

Find the length of the arc of the curve from point to point Q

Find the length of the arc of the curve from point to point Q

Graph the curve and visually estimate its length. Then find its exact length.

Graph the curve and visually estimate its length. Then find its exact length.

Use Simpsons Rule with to estimate the arc length of the curve. Compare your answer with the value of the integral produced by your calculator

Use Simpsons Rule with to estimate the arc length of the curve. Compare your answer with the value of the integral produced by your calculator

Use Simpsons Rule with to estimate the arc length of the curve. Compare your answer with the value of the integral produced by your calculator

Use Simpsons Rule with to estimate the arc length of the curve. Compare your answer with the value of the integral produced by your calculator

(a) Graph the curve , . (b) Compute the lengths of inscribed polygons with , , and sides. (Divide the interval into equal subintervals.) Illustrate by sketching these polygons (as in Figure 6). (c) Set up an integral for the length of the curve. (d) Use your calculator to find the length of the curve to four decimal places. Compare with the approximations in part (b).

Repeat Exercise 27 for the curve

Use either a computer algebra system or a table of integrals to find the exact length of the arc of the curve that lies between the points and .

Use either a computer algebra system or a table of integrals to find the exact length of the arc of the curve that lies between the points and . If your CAS has trouble evaluating the integral, make a substitution that changes the integral into one that the CAS can evaluate.

Sketch the curve with equation and use symmetry to find its length

(a) Sketch the curve . (b) Use Formulas 3 and 4 to set up two integrals for the arc length from to . Observe that one of these is an improper integral and evaluate both of them. (c) Find the length of the arc of this curve from to .

Find the arc length function for the curve with starting point

(a) Graph the curve , . (b) Find the arc length function for this curve with starting point . (c) Graph the arc length function

Find the arc length function for the curve with starting point

A steady wind blows a kite due west. The kites height above ground from horizontal position to is given by . Find the distance traveled by the kite.

A hawk flying at at an altitude of 180 m accidentally drops its prey. The parabolic trajectory of the falling prey is described by the equation until it hits the ground, where is its height above the ground and is the horizontal distance traveled in meters. Calculate the distance traveled by the prey from the time it is dropped until the time it hits the ground. Express your answer correct to the nearest tenth of a meter.

The Gateway Arch in St. Louis (see the photo on page 256) was constructed using the equation for the central curve of the arch, where and are measured in meters and . Set up an integral for the length of the arch and use your calculator to estimate the length correct to the nearest meter.

A manufacturer of corrugated metal roofing wants to produce panels that are 28 in. wide and 2 in. thick by processing flat sheets of metal as shown in the figure. The profile of the roofing takes the shape of a sine wave. Verify that the sine curve has equation and find the width of a flat metal sheet that is needed to make a 28-inch panel. (Use your calculator to evaluate the integral correct to four significant digits.)

a) The figure shows a telephone wire hanging between two poles at and . It takes the shape of a catenary with equation . Find the length of the wire. ; (b) Suppose two telephone poles are 50 ft apart and the length of the wire between the poles is 51 ft. If the lowest point of the wire must be 20 ft above the ground, how high up on each pole should the wire be attached?

Find the length of the curve ; The curves with equations , , , , . . . , are called fat circles. Graph the curves with , , , , and to see why. Set up an integral for the length of the fat circle with . Without attempting to evaluate this integral, state the value of .

The curve , , is an arc of the circle . Find the area of the surface obtained by rotating this arc about the -axis. (The surface is a portion of a sphere of radius 2. See Figure 6.)

The arc of the parabola from to is rotated about the -axis. Find the area of the resulting surface.

Find the area of the surface generated by rotating the curve , , about the -axis

Set up, but do not evaluate, an integral for the area of the n 10 surface obtained by rotating the curve about (a) the -axis and (b) the -axis.

Set up, but do not evaluate, an integral for the area of the n 10 surface obtained by rotating the curve about (a) the -axis and (b) the -axis.

Set up, but do not evaluate, an integral for the area of the n 10 surface obtained by rotating the curve about (a) the -axis and (b) the -axis.

Set up, but do not evaluate, an integral for the area of the n 10 surface obtained by rotating the curve about (a) the -axis and (b) the -axis.

Find the area of the surface obtained by rotating the curve about the -axis

Find the area of the surface obtained by rotating the curve about the -axis

Find the area of the surface obtained by rotating the curve about the -axis

Find the area of the surface obtained by rotating the curve about the -axis

Find the area of the surface obtained by rotating the curve about the -axis

Find the area of the surface obtained by rotating the curve about the -axis

Find the area of the surface obtained by rotating the curve about the -axis

Find the area of the surface obtained by rotating the curve about the -axis

The given curve is rotated about the -axis. Find the area of the resulting surface.

The given curve is rotated about the -axis. Find the area of the resulting surface.

The given curve is rotated about the -axis. Find the area of the resulting surface.

The given curve is rotated about the -axis. Find the area of the resulting surface.

Use Simpsons Rule with to approximate the area of the surface obtained by rotating the curve about the -axis. Compare your answer with the value of the integral produced by your calculator

Use Simpsons Rule with to approximate the area of the surface obtained by rotating the curve about the -axis. Compare your answer with the value of the integral produced by your calculator

Use Simpsons Rule with to approximate the area of the surface obtained by rotating the curve about the -axis. Compare your answer with the value of the integral produced by your calculator

Use Simpsons Rule with to approximate the area of the surface obtained by rotating the curve about the -axis. Compare your answer with the value of the integral produced by your calculator

Use either a CAS or a table of integrals to find the exact area of the surface obtained by rotating the given curve about the -axis.

Use either a CAS or a table of integrals to find the exact area of the surface obtained by rotating the given curve about the -axis.

Use a CAS to find the exact area of the surface obtained by rotating the curve about the -axis. If your CAS has trouble evaluating the integral, express the surface area as an integral in the other variable

Use a CAS to find the exact area of the surface obtained by rotating the curve about the -axis. If your CAS has trouble evaluating the integral, express the surface area as an integral in the other variable

If the region is rotated about the -axis, the volume of the resulting solid is finite (see Exercise 63 in Section 7.8). Show that the surface area is infinite. (The surface is shown in the figure and is known as Gabriels horn.)

If the infinite curve , , is rotated about the CAS -axis, find the area of the resulting surface

(a) If , find the area of the surface generated by rotating the loop of the curve about the -axis. (b) Find the surface area if the loop is rotated about the -axis

A group of engineers is building a parabolic satellite dish whose shape will be formed by rotating the curve about the -axis. If the dish is to have a 10-ft diameter and a maximum depth of 2 ft, find the value of and the surface area of the dish.

a) The ellipse is rotated about the -axis to form a surface called an ellipsoid, or prolate spheroid. Find the surface area of this ellipsoid. (b) If the ellipse in part (a) is rotated about its minor axis (the -axis), the resulting ellipsoid is called an oblate spheroid. Find the surface area of this ellipsoid

Find the surface area of the torus in Exercise 63 in Section 6.2

If the curve , , is rotated about the horizontal line , where , find a formula for the area of the resulting surface.

Use the result of Exercise 31 to set up an integral to find the area of the surface generated by rotating the curve , , about the line . Then use a CAS to evaluate the integral

Find the area of the surface obtained by rotating the circle about the line y=r

Show that the surface area of a zone of a sphere that lies between two parallel planes is , where is the diameter of the sphere and is the distance between the planes. (Notice that depends only on the distance between the planes and not on their location, provided that both planes intersect the sphere.)

Formula 4 is valid only when . Show that when is not necessarily positive, the formula for surface area becomes S y b a 2 f x s1 f x 2 dx f

Let be the length of the curve , , where is positive and has a continuous derivative. Let be the surface area generated by rotating the curve about the -axis. If is a positive constant, define and let be the corresponding surface area generated by the curve y tx, . Express in terms of and .

A dam has the shape of the trapezoid shown in Figure 2. The height is 20 m, and the width is 50 m at the top and 30 m at the bottom. Find the force on the dam due to hydrostatic pressure if the water level is 4 m from the top of the dam.

Find the hydrostatic force on one end of a cylindrical drum with radius 3 ft if the drum is submerged in water 10 ft deep

Find the moments and center of mass of the system of objects that have masses 3, 4, and 8 at the points , , and , respectively

Find the center of mass of a semicircular plate of radius r

Find the centroid of the region bounded by the curves , , , an

Find the centroid of the region bounded by the line and the parabola y x

A torus is formed by rotating a circle of radius about a line in the plane of the circle that is a distance from the center of the circle. Find the volume of the torus.

An aquarium 5 ft long, 2 ft wide, and 3 ft deep is full of water. Find (a) the hydrostatic pressure on the bottom of the aquarium, (b) the hydrostatic force on the bottom, and (c) the hydrostatic force on one end of the aquarium.

A tank is 8 m long, 4 m wide, 2 m high, and contains kerosene with density to a depth of 1.5 m. Find (a) the hydrostatic pressure on the bottom of the tank, (b) the hydrostatic force on the bottom, and (c) the hydrostatic force on one end of the tank.

A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum. Then express the force as an integral and evaluate it.

A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum. Then express the force as an integral and evaluate it.

A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum. Then express the force as an integral and evaluate it.

A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum. Then express the force as an integral and evaluate it.

A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum. Then express the force as an integral and evaluate it.

A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum. Then express the force as an integral and evaluate it.

A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum. Then express the force as an integral and evaluate it.

A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum. Then express the force as an integral and evaluate it.

A vertical plate is submerged (or partially submerged) in water and has the indicated shape. Explain how to approximate the hydrostatic force against one side of the plate by a Riemann sum. Then express the force as an integral and evaluate it.

A large tank is designed with ends in the shape of the region between the curves and , measured in feet. Find the hydrostatic force on one end of the tank if it is filled to a depth of 8 ft with gasoline. (Assume the gasolines density is .)

A trough is filled with a liquid of density 840 kgm . The ends of the trough are equilateral triangles with sides 8 m long and vertex at the bottom. Find the hydrostatic force on one end of the trough.

A vertical dam has a semicircular gate as shown in the figure. Find the hydrostatic force against the gate.

A cube with 20-cm-long sides is sitting on the bottom of an aquarium in which the water is one meter deep. Estimate the hydrostatic force on (a) the top of the cube and (b) one of the sides of the cube.

A dam is inclined at an angle of from the vertical and has the shape of an isosceles trapezoid 100 ft wide at the top and 50 ft wide at the bottom and with a slant height of 70 ft. Find the hydrostatic force on the dam when it is full of water

A swimming pool is 20 ft wide and 40 ft long and its bottom is an inclined plane, the shallow end having a depth of 3 ft and the deep end, 9 ft. If the pool is full of water, estimate the hydrostatic force on (a) the shallow end, (b) the deep end, (c) one of the sides, and (d) the bottom of the pool.

Suppose that a plate is immersed vertically in a fluid with density and the width of the plate is at a depth of meters beneath the surface of the fluid. If the top of the plate is at depth and the bottom is at depth , show that the hydrostatic force on one side of the plate is F y b a txwxdx a

A vertical, irregularly shaped plate is submerged in water. The table shows measurements of its width, taken at the indicated depths. Use Simpsons Rule to estimate the force of the water against the plate.

(a) Use the formula of Exercise 18 to show that where is the -coordinate of the centroid of the plate and is its area. This equation shows that the hydrostatic force against a vertical plane region is the same as if the region were horizontal at the depth of the centroid of the region. (b) Use the result of part (a) to give another solution to Exercise 10.

Point-masses are located on the -axis as shown. Find the moment of the system about the origin and the center of mass

Point-masses are located on the -axis as shown. Find the moment of the system about the origin and the center of mass

The masses are located at the points . Find the moments and and the center of mass of the system

The masses are located at the points . Find the moments and and the center of mass of the system

Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid.

Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid.

Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid.

Sketch the region bounded by the curves, and visually estimate the location of the centroid. Then find the exact coordinates of the centroid.

Find the centroid of the region bounded by the given curves

Find the centroid of the region bounded by the given curves

Find the centroid of the region bounded by the given curves

Find the centroid of the region bounded by the given curves

Find the centroid of the region bounded by the given curves

Calculate the moments and and the center of mass of a lamina with the given density and shape.

Calculate the moments and and the center of mass of a lamina with the given density and shape.

Use Simpsons Rule to estimate the centroid of the region shown

Find the centroid of the region bounded by the curves and , , to three decimal places. Sketch the region and plot the centroid to see if your answer is reasonable

Use a graph to find approximate -coordinates of the points of intersection of the curves and . Then find (approximately) the centroid of the region bounded by these curves

Prove that the centroid of any triangle is located at the point of intersection of the medians. [Hints: Place the axes so that the vertices are , , and . Recall that a median is a line segment from a vertex to the midpoint of the opposite side. Recall also that the medians intersect at a point twothirds of the way from each vertex (along the median) to the opposite side.]

Find the centroid of the region shown, not by integration, but by locating the centroids of the rectangles and triangles (from Exercise 39) and using additivity of moments.

Find the centroid of the region shown, not by integration, but by locating the centroids of the rectangles and triangles (from Exercise 39) and using additivity of moments.

A rectangle with sides and is divided into two parts and by an arc of a parabola that has its vertex at one corner of and passes through the opposite corner. Find the centroids of both and .

If is the -coordinate of the centroid of the region that lies under the graph of a continuous function , where , show that y b a cx d f x dx cx d y b a f x dx f a

Use the Theorem of Pappus to find the volume of the given solid A sphere of radius r

Use the Theorem of Pappus to find the volume of the given solid A cone with height and base radius r

Use the Theorem of Pappus to find the volume of the given solid The solid obtained by rotating the triangle with vertices , , and about the -axis

Prove Formulas 9.

Let be the region that lies between the curves and , , where and are integers with . (a) Sketch the region . (b) Find the coordinates of the centroid of . (c) Try to find values of and such that the centroid lies outside

The demand for a product, in dollars, is p 1200 0.2x 0.0001x

A 5-mg bolus of dye is injected into a right atrium. The concentration of the dye (in milligrams per liter) is measured in the aorta at one-second intervals as shown in the chart. Estimate the cardiac output.

The marginal cost function was defined to be the p 20 0.05x derivative of the cost function. (See Sections 3.7 and 4.7.) If the marginal cost of maufacturing meters of a fabric is (measured in dollars per meter) and the fixed start-up cost is , use the Net Change Theorem to find the cost of producing the first 2000 units.

The marginal revenue from the sale of units of a product is . If the revenue from the sale of the first 1000 units is $12,400, find the revenue from the sale of the first 5000 units

The demand function for a certain commodity is . Find the consumer surplus when the sales level is 300. Illustrate by drawing the demand curve and identifying the consumer surplus as an area

A demand curve is given by . Find the consumer surplus when the selling price is .

The supply function for a commodity gives the relation between the selling price and the number of units that manufacturers will produce at that price. For a higher price, manufacturers will produce more units, so is an increasing function of . Let be the amount of the commodity currently produced and let be the current price. Some producers would be willing to make and sell the commodity for a lower selling price and are therefore receiving more than their minimal price. The excess is called the producer surplus. An argument similar to that for consumer surplus shows that the surplus is given by the integral Calculate the producer surplus for the supply function at the sales level . Illustrate by drawing the supply curve and identifying the producer surplus as an area.

If a supply curve is modeled by the equation , find the producer surplus when the selling price is $400.

For a given commodity and pure competition, the number of units produced and the price per unit are determined as the coordinates of the point of intersection of the supply and demand curves. Given the demand curve and the supply curve , find the consumer surplus and the producer surplus. Illustrate by sketching the supply and demand curves and identifying the surpluses as areas

A company modeled the demand curve for its product (in dollars) by the equation Use a graph to estimate the sales level when the selling price is $16. Then find (approximately) the consumer surplus for this sales level.

A movie theater has been charging $7.50 per person and selling about 400 tickets on a typical weeknight. After surveying their customers, the theater estimates that for every 50 cents that they lower the price, the number of moviegoers will increase by 35 per night. Find the demand function and calculate the consumer surplus when the tickets are priced at $6.00.

If the amount of capital that a company has at time is , then the derivative, , is called the net investment flow. Suppose that the net investment flow is million dollars per year (where is measured in years). Find the increase in capital (the capital formation) from the fourth year to the eighth year

If revenue flows into a company at a rate of , where is measured in years and is measured in dollars per year, find the total revenue obtained in the first four years.

If revenue flows into a company at a rate of , where is measured in years and is measured in dollars per year, find the total revenue obtained in the first four years.

A hot, wet summer is causing a mosquito population explosion in a lake resort area. The number of mosquitos is increasing at an estimated rate of per week (where is measured in weeks). By how much does the mosquito population increase between the fifth and ninth weeks of summer?

Use Poiseuilles Law to calculate the rate of flow in a small human artery where we can take , cm, cm, and dynescm .

High blood pressure results from constriction of the arteries. To maintain a normal flow rate (flux), the heart has to pump harder, thus increasing the blood pressure. Use Poiseuilles Law to show that if and are normal values of the radius and pressure in an artery and the constricted values are and , then for the flux to remain constant, and are related by the equation Deduce that if the radius of an artery is reduced to threefourths of its former value, then the pressure is more than tripled.

The dye dilution method is used to measure cardiac output with 6 mg of dye. The dye concentrations, in , are modeled by , , where is measured in seconds. Find the cardiac output

After an 8-mg injection of dye, the readings of dye concentration, in , at two-second intervals are as shown in the table. Use Simpsons Rule to estimate the cardiac output.

The graph of the concentration function is shown after a 7-mg injection of dye into a heart. Use Simpsons Rule to estimate the cardiac output.

Let for and for all other values of . (a) Verify that is a probability density function. (b) Find . P4 X 8 f

Phenomena such as waiting times and equipment failure times are commonly modeled by exponentially decreasing probability density functions. Find the exact form of such a function.

Find the mean of the exponential distribution of Example 2:ft 0 cect if t 0 if t 0 5

Suppose the average waiting time for a customers call to be answered by a company representative is five minutes. (a) Find the probability that a call is answered during the first minute. (b) Find the probability that a customer waits more than five minutes to be answered.

Intelligence Quotient (IQ) scores are distributed normally with mean 100 and standard deviation 15. (Figure 6 shows the corresponding probability density function.) (a) What percentage of the population has an IQ score between 85 and 115? (b) What percentage of the population has an IQ above 140?

Let be the probability density function for the lifetime of a  manufacturers highest quality car tire, where is measured in miles. Explain the meaning of each integral. (a) (b

Let be the probability density function for the time it takes you to drive to school in the morning, where is measured in minutes. Express the following probabilities as integrals. (a) The probability that you drive to school in less than 15 minutes (b) The probability that it takes you more than half an hour to get to school

Let for and for all other values of . (a) Verify that is a probability density function. (b) Find

Let if and if . (a) Verify that is a probability density function. (b) Find . P

Let . (a) For what value of is a probability density function? (b) For that value of , find

Let if and if or . (a) For what value of is a probability density function? (b) For that value of , find . (c) Find the mean.

A spinner from a board game randomly indicates a real number between 0 and 10. The spinner is fair in the sense that it indicates a number in a given interval with the same probability as it indicates a number in any other interval of the same length. (a) Explain why the function is a probability density function for the spinners values. (b) What does your intuition tell you about the value of the mean? Check your guess by evaluating an integral.

(a) Explain why the function whose graph is shown is a probability density function. (b) Use the graph to find the following probabilities: (i) (ii) (c) Calculate the mean

Show that the median waiting time for a phone call to the company described in Example 4 is about 3.5 minutes.

(a) A type of lightbulb is labeled as having an average lifetime of 1000 hours. Its reasonable to model the probability of failure of these bulbs by an exponential density function with mean . Use this model to find the probability that a bulb (i) fails within the first 200 hours, (ii) burns for more than 800 hours. (b) What is the median lifetime of these lightbulbs?

The manager of a fast-food restaurant determines that the average time that her customers wait for service is 2.5 minutes. (a) Find the probability that a customer has to wait more than 4 minutes. (b) Find the probability that a customer is served within the first 2 minutes. (c) The manager wants to advertise that anybody who isnt served within a certain number of minutes gets a free hamburger. But she doesnt want to give away free hamburgers to more than 2% of her customers. What should the advertisement say?

According to the National Health Survey, the heights of adult males in the United States are normally distributed with mean 69.0 inches and standard deviation 2.8 inches. (a) What is the probability that an adult male chosen at random is between 65 inches and 73 inches tall? (b) What percentage of the adult male population is more than 6 feet tall?

The Garbage Project at the University of Arizona reports that the amount of paper discarded by households per week is normally distributed with mean 9.4 lb and standard deviation 4.2 lb. What percentage of households throw out at least 10 lb of paper a week?

Boxes are labeled as containing 500 g of cereal. The machine filling the boxes produces weights that are normally distributed with standard deviation 12 g. (a) If the target weight is 500 g, what is the probability that the machine produces a box with less than 480 g of cereal? (b) Suppose a law states that no more than 5% of a manufacturers cereal boxes can contain less than the stated weight of 500 g. At what target weight should the manufacturer set its filling machine?

The speeds of vehicles on a highway with speed limit are normally distributed with mean and standard deviation . (a) What is the probability that a randomly chosen vehicle is traveling at a legal speed? (b) If police are instructed to ticket motorists driving or more, what percentage of motorists are targeted?

Show that the probability density function for a normally distributed random variable has inflection points at x

For any normal distribution, find the probability that the random variable lies within two standard deviations of the mean

The standard deviation for a random variable with probability density function and mean is defined by Find the standard deviation for an exponential density function with mean

The hydrogen atom is composed of one proton in the nucleus and one electron, which moves about the nucleus. In the quantum theory of atomic structure, it is assumed that the electron does not move in a well-defined orbit. Instead, it occupies a state known as an orbital, which may be thought of as a cloud of negative charge surrounding the nucleus. At the state of lowest energy, called the ground state, or 1s-orbital, the shape of this cloud is assumed to be a sphere centered at the nucleus. This sphere is described in terms of the probability density function where is the Bohr radius . The integral gives the probability that the electron will be found within the sphere of radius meters centered at the nucleus. (a) Verify that is a probability density function. (b) Find . For what value of does have its maximum value? ; (c) Graph the density function. (d) Find the probability that the electron will be within the sphere of radius centered at the nucleus. (e) Calculate the mean distance of the electron from the nucleus in the ground state of the hydrogen atom.

a) How is the length of a curve defined? px (b) Write an expression for the length of a smooth curve given by , . (c) What if is given as a function of ?

(a) Write an expression for the surface area of the surface obtained by rotating the curve , , about the -axis. (b) What if is given as a function of ? (c) What if the curve is rotated about the -axis?

Describe how we can find the hydrostatic force against a vertical wall submersed in a fluid.

a) What is the physical significance of the center of mass of a thin plate? (b) If the plate lies between and , where , write expressions for the coordinates of the center of mass

What does the Theorem of Pappus say?

Given a demand function , explain what is meant by the consumer surplus when the amount of a commodity currently available is and the current selling price is . Illustrate with a sketch.

a) What is the cardiac output of the heart? (b) Explain how the cardiac output can be measured by the dye dilution method.

What is a probability density function? What properties does such a function have?

Suppose is the probability density function for the weight of a female college student, where is measured in pounds. (a) What is the meaning of the integral ? (b) Write an expression for the mean of this density function. (c) How can we find the median of this density function?

What is a normal distribution? What is the significance of the standard deviation?

Find the length of the curve.

Find the length of the curve.

(a) Find the length of the curve (b) Find the area of the surface obtained by rotating the curve in part (a) about the -axis.

(a) The curve , , is rotated about the -axis. Find the area of the resulting surface. (b) Find the area of the surface obtained by rotating the curve in part (a) about the -axis.

Use Simpsons Rule with to estimate the length of the curve ,

Use Simpsons Rule with to estimate the area of the surface obtained by rotating the curve in Exercise 5 about the x-axis.

Find the length of the curve y y 1 x 16 x 1 sst 1 dt EX

Find the area of the surface obtained by rotating the curve in Exercise 7 about the -axis

A gate in an irrigation canal is constructed in the form of a trapezoid 3 ft wide at the bottom, 5 ft wide at the top, and 2 ft high. It is placed vertically in the canal so that the water just covers the gate. Find the hydrostatic force on one side of the gate.

A trough is filled with water and its vertical ends have the shape of the parabolic region in the figure. Find the hydrostatic force on one end of the trough.

Find the centroid of the region bounded by the given curves

Find the centroid of the region bounded by the given curves

Find the centroid of the region shown

Find the centroid of the region shown

Find the volume obtained when the circle of radius 1 with center is rotated about the -axis.

Use the Theorem of Pappus and the fact that the volume of a sphere of radius is to find the centroid of the semicircular region bounded by the curve and the -axis.

The demand function for a commodity is given by Find the consumer surplus when the sales level is 100

After a 6-mg injection of dye into a heart, the readings of dye concentration at two-second intervals are as shown in the table. Use Simpsons Rule to estimate the cardiac output

(a) Explain why the function is a probability density function. (b) Find . (c) Calculate the mean. Is the value what you would expect?

Lengths of human pregnancies are normally distributed with mean 268 days and standard deviation 15 days. What percentage of pregnancies last between 250 days and 280 days?

The length of time spent waiting in line at a certain bank is modeled by an exponential density function with mean 8 minutes. (a) What is the probability that a customer is served in the first 3 minutes? (b) What is the probability that a customer has to wait more than 10 minutes? (c) What is the median waiting time?

Find the area of the region

Find the centroid of the region enclosed by the loop of the curve y 2 x 3 x 4

If a sphere of radius is sliced by a plane whose distance from the center of the sphere is , then the sphere is divided into two pieces called segments of one base. The corresponding surfaces are called spherical zones of one base. (a) Determine the surface areas of the two spherical zones indicated in the figure. (b) Determine the approximate area of the Arctic Ocean by assuming that it is approximately circular in shape, with center at the North Pole and circumference at north latitude. Use mi for the radius of the earth. (c) A sphere of radius is inscribed in a right circular cylinder of radius . Two planes perpendicular to the central axis of the cylinder and a distance apart cut off a spherical zone of two bases on the sphere. Show that the surface area of the spherical zone equals the surface area of the region that the two planes cut off on the cylinder. (d) The Torrid Zone is the region on the surface of the earth that is between the Tropic of Cancer ( north latitude) and the Tropic of Capricorn ( south latitude). What is the area of the Torrid Zone?

a) Show that an observer at height above the north pole of a sphere of radius can see a part of the sphere that has area (b) Two spheres with radii and are placed so that the distance between their centers is , where . Where should a light be placed on the line joining the centers of the spheres in order to illuminate the largest total surface?

Suppose that the density of seawater, , varies with the depth below the surface. (a) Show that the hydrostatic pressure is governed by the differential equation where is the acceleration due to gravity. Let and be the pressure and density at . Express the pressure at depth as an integral. (b) Suppose the density of seawater at depth is given by , where is a positive constant. Find the total force, expressed as an integral, exerted on a vertical circular porthole of radius whose center is located at a distance below the surface.

The figure shows a semicircle with radius 1, horizontal diameter , and tangent lines at and . At what height above the diameter should the horizontal line be placed so as to minimize the shaded area?

Let be a pyramid with a square base of side and suppose that is a sphere with its center on the base of and is tangent to all eight edges of . Find the height of . Then find the volume of the intersection of and .

Consider a flat metal plate to be placed vertically under water with its top 2 m below the surface of the water. Determine a shape for the plate so that if the plate is divided into any number of horizontal strips of equal height, the hydrostatic force on each strip is the same

A uniform disk with radius 1 m is to be cut by a line so that the center of mass of the smaller piece lies halfway along a radius. How close to the center of the disk should the cut be made? (Express your answer correct to two decimal places.)

A triangle with area is cut from a corner of a square with side 10 cm, as shown in the figure. If the centroid of the remaining region is 4 cm from the right side of the square, how far is it from the bottom of the square?

In a famous 18th-century problem, known as Buffons needle problem, a needle of length is dropped onto a flat surface (for example, a table) on which parallel lines units apart, , have been drawn. The problem is to determine the probability that the needle will come to rest intersecting one of the lines. Assume that the lines run east-west, parallel to the -axis in a rectangular coordinate system (as in the figure). Let be the distance from the southern end of the needle to the nearest line to the north. (If the needles southern end lies on a line, let . If the needle happens to lie east-west, let the western end be the southern end.) Let be the angle that the needle makes with a ray extending eastward from the southern end. Then and . Note that the needle intersects one of the lines only when . The total set of possibilities for the needle can be identified with the rectangular region , , and the proportion of times that the needle intersects a line is the ratio This ratio is the probability that the needle intersects a line. Find the probability that the needle will intersect a line if . What if

If the needle in Problem 11 has length , its possible for the needle to intersect more than one line. (a) If , find the probability that a needle of length 7 will intersect at least one line. [Hint: Proceed as in Problem 11. Define as before; then the total set of possibilities for the needle can be identified with the same rectangular region , . What portion of the rectangle corresponds to the needle intersecting a line?] (b) If , find the probability that a needle of length 7 will intersect two lines. (c) If , find a general formula for the probability that the needle intersects three lines.