?Let \(\mathscr{R}\) be the region in the first quadrant bounded by the curves

(a) Draw two typical curves \(y=f(x)\) and \(y=g(x)\), where \(f(x) \geqslant g(x)\) for \(a \leqslant x \leqslant b\) Show how to approximate the area between these curves by a Riemann sum and sketch the corresponding approximating rectangles. Then write an expression for the exact area. (b) Explain how the situation changes if the curves have equations \(x=f(y)\) and \(x=g(y)\), where \(f(y) \geqslant g(y)\) for \(c \leqslant y \leqslant d\).

Suppose that Sue runs faster than Kathy throughout a 1500 -meter race. What is the physical meaning of the area between their velocity curves for the first minute of the race?

(a) Suppose \(S\) is a solid with known cross-sectional areas. Explain how to approximate the volume of \(S\) by a Riemann sum. Then write an expression for the exact volume. (b) If \(S\) is a solid of revolution, how do you find the cross-sectional areas? Equation Transcription: Text Transcription: S

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. Let R be the region shown. If R is revolved about the y-axis, then the volume of the resulting solid is \(V=\int_{a}^{b} 2 \pi x f(x) d x\).

Suppose that you push a book across a 6-meter-long table by exerting a force f(x) at each point from x = 0 to x = 6. What does \(\int_{0}^{6} f(x) d x\) represent? If f(x) is measured in newtons, what are the units for the integral?

(a) What is the average value of a function \(f\) on an interval \([a,b]\)? (b) What does the Mean Value Theorem for Integrals say? What is its geometric interpretation? Equation Transcription: Text Transcription: f [a,b]

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. The area between the curves \(y=f(x)\) and \(y=g(x)\) for \(a \leqslant x \leqslant b \text { is } A=\int_{a}^{b}[f(x)-g(x)] d x\).

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. A cube is a solid of revolution.

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. If the region bounded by the curves \(y=\sqrt{x}\) and \(y=x\) is revolved about the x-axis, then the volume of the resulting solid is \(V=\int_{0}^{1} \pi(\sqrt{x}-x)^{2} \ d x\). Equation Transcription: y = y = x V = ? ( - x)2 dx Text Transcription: y = sqrt x y = x V = integral_0 ^1 pi (sqrt x - x)^2 dx

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. Let be the region shown. If is revolved about the -axis, then the volume of the resulting solid is .

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. Let \(\mathscr{R}\) be the region shown. If \(\mathscr{R}\) is revolved about the \(x\)-axis, then the volume of the resulting solid is \(V=\int_{a}^{b} \ \pi[f(x)]^{2} \ d x\). Equation Transcription: R x V = ? [f(x)]2 dx Text Transcription: R x V=integral_a ^b pi[f(x)]^2 dx

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. Let be the region shown. If is revolved about the x -axis, then vertical cross-sections perpendicular to the x- axis of the resulting solid are disks.

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. \(y=2x,\ \ \ y=x^2\); about the \(x\)-axis Equation Transcription: Text Transcription: y = 2x, y = x^2 x

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. \(x=1+y^2,\ \ \ y=x-3\); about the \(y\)-axis Equation Transcription: Text Transcription: x = 1 + y^2, y = x - 3 y

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. \(x=0,\ \ \ x=9-y^2\); about \(x=-1\) Equation Transcription: Text Transcription: x = 0 + y^2, y = x - 3 x = -1

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. \(y=x^2+1,\ \ \ y=9-x^2\); about \(y=-1\) Equation Transcription: Text Transcription: y = x^2 + 1, y = 9 - x^2 y = -1

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. \(x^2-y^2=a^2, x=a+h\) (where a>0, h>0 ); about the y-axis

Find the area of the region bounded by the given curves.

Find the area of the region bounded by the given curves. \(y=\sqrt{x}, \quad y=-\sqrt[3]{x}, \quad y=x-2\)

Find the area of the region bounded by the given curves. \(y=1-2 x^2, \quad y=|x|\)

Find the area of the region bounded by the given curves.

Find the area of the region bounded by the given curves. \(y=\sin(\pi x/2),\quad\ \ \ y=x^2-2x\)

Find the area of the region bounded by the given curves.

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis.

Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. \(y=\tan\ x,\ \ y=x,\ \ x=\pi/3\); about the \(y\)-axis

Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. \(y=\cos ^{2} x, \ \ |x| \leqslant \pi / 2, \ \ y=\frac{1}{4}\) about \(x=\pi / 2\) Equation Transcription: ? Text Transcription: y = cos^2x, |x| ? pi/2, y=1/4 x = pi/2

Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified axis. \(y=\ln x,\ \ y=0,\ \ x=4\); about \(x=-1\) Equation Transcription: Text Transcription: y = ln x, y=0, x = 4 x = -1

The region bounded by the given curves is rotated about the specified axis. Find the volume of the solid using (a) \(x\) as the variable of integration and (b) \(y\) as the variable of integration. \(y=x^3,\ \ y=3x^2\); about \(x=-1\) Equation Transcription: Text Transcription: x y y = x3^, y = 3x^2 x = -1

The region bounded by the given curves is rotated about the specified axis. Find the volume of the solid using (a) \(x\) as the variable of integration and (b) \(y\) as the variable of integration. \(y=\sqrt{x},\ \ y=x^2\); about \(y=3\) Equation Transcription: Text Transcription: x y y = x, y = x^2 y = 3

Find the volumes of the solids obtained by rotating the region bounded by the curves y = x and \(y = x^2\) about the following lines. (a) The x-axis (b) The y-axis (c) y = 2

Let \(\mathscr{R}\) be the region in the first quadrant bounded by the curves \(y=x^{3}\) and \(y=2 x-x^{2}\). Calculate the following quantities. (a) The area of \(\mathscr{R}\) (b) The volume obtained by rotating \(\mathscr{R}\) about the \(x\)-axis (c) The volume obtained by rotating \(\mathscr{R}\) about the \(y\)-axis Equation Transcription: ? ? ? ? Text Transcription: ? y = x^3 y = 2x - x^2 ? ? x ? y

Let \(\mathscr{R}\) be the region bounded by the curves \(y=\tan \left(x^{2}\right)\), \(x=1\), and \(y=0\). Use the Midpoint Rule with \(n=4\) to estimate the following quantities. (a) The area of \(\mathscr{R}\) (b) The volume obtained by rotating 5 about the \(x\)-axis Equation Transcription: ? ? ? Text Transcription: ? y = tan(x^2) x = 1 y = 0 n =4 ? ? x

Let \(\mathscr{R}\) be the region bounded by the curves \(y=1-x^{2}\) and \(y=x^{6}-x+1\). Estimate the following quantities. (a) The \(x\)-coordinates of the points of intersection of the curves (b) The area of \(\mathscr{R}\) (c) The volume generated when \(\mathscr{R}\) is rotated about the \(x\)-axis (d) The volume generated when \(\mathscr{R}\) is rotated about the \(y\)-axis Equation Transcription: ? ? ? ? Text Transcription: ? y = 1 - x^2 y = x^6 - x + 1 x ? ? x ? y

Each integral represents the volume of a solid. Describe the solid. \(\int_0^{\pi/2}\ 2\pi x\ \cos\ x\ dx\) Equation Transcription: Text Transcription: Int_0 ^ pi/2 2 pi x cos x dx

Each integral represents the volume of a solid. Describe the solid. \(\int_0^{\pi/2}\ 2\pi\ \cos^2x\ dx\) Equation Transcription: Text Transcription: int_0 ^pi/2 2pi cos^2 x dx

Each integral represents the volume of a solid. Describe the solid. \(\int_0^\pi \pi(2-\sin x)^2 d x\)

Each integral represents the volume of a solid. Describe the solid. \(\int_0^4 2 \pi(6-y)\left(4 y-y^2\right) d y\)

The base of a solid is a circular disk with radius \(3\). Find the volume of the solid if parallel cross-sections perpendicular to the base are isosceles right triangles with hypotenuse lying along the base. Equation Transcription: Text Transcription: 3

The base of a solid is the region bounded by the parabolas \(y=x^{2}\) and \(y=2-x^{2}\). Find the volume of the solid if the cross-sections perpendicular to the \(x\)-axis are squares with one side lying along the base. Equation Transcription: Text Transcription: y = x^2 y = 2 - x^2 x

The height of a monument is \(20 \mathrm{~m}\). A horizontal cross section at a distance x meters from the top is an equilateral triangle with side \(\frac{1}{4} x\) meters. Find the volume of the monument.

(a) The base of a solid is a square with vertices located at \((1,\ 0)\), \((0,\ 1)\), \((-1,\ 0)\), and \((0,\ -1)\). Each cross-section perpendicular to the \(x\)-axis is a semicircle. Find the volume of the solid. (b) Show that the volume of the solid of part (a) can be computed more simply by first cutting the solid and rearranging it to form a cone. Equation Transcription: Text Transcription: (1, 0) (0, 1) (-1, 0) (0, -1) x

A force of 30 N is required to maintain a spring stretched from its natural length of 12 cm to a length of 15 cm. How much work is done in stretching the spring from 12 cm to 20 cm?

A 1600-lb elevator is suspended by a 200-ft cable that weighs 10 lb/ft. How much work is required to raise the elevator from the basement to the third floor, a distance of 30 ft?

A tank full of water has the shape of a paraboloid of revolution as shown in the figure; that is, its shape is obtained by rotating a parabola about a vertical axis. (a) If its height is \(4 \ ft\) and the radius at the top is \(4 \ ft\), find the work required to pump the water out of the tank. (b) After \(4000 \ ft-lb\) of work has been done, what is the depth of the water remaining in the tank? Equation Transcription: Text Transcription: 4 ft 4 ft 4000 ft-lb

A steel tank has the shape of a circular cylinder oriented vertically with diameter \(4 \ m\) and height \(5 \ m\). The tank is currently filled to a level of \(3 \ m\) with cooking oil that has a density of \(920\mathrm{\ kg}/\mathrm{m}^3\). Compute the work required to pump the oil out through a \(1-m\) spout at the top of the tank. Equation Transcription: Text Transcription: 4 m 5 m 3 m 920 kg/m^3 1-m

Find the average value of the function \(f(t)=\sec ^2 t\) on the interval \([0, \pi / 4]\)

(a) Find the average value of the function \(f(x)=1 / \sqrt{x}\) on the interval \([1,\ 4]\). (b) Find the value c guaranteed by the Mean Value Theorem for Integrals such that \(f_{\mathrm{avg}}=f(c)\). (c) Sketch the graph of \(f\) on \([1,\ 4]\) and a rectangle with base \([1,\ 4]\) whose area is the same as the area under the graph of \(f\). Equation Transcription: Text Transcription: f(x) = 1/ sqrt x [1, 4] f_avg = f(c) f [1, 4] [1, 4] f

Let \(\mathscr{R}_{1}\) be the region bounded by \(y=x^{2}\), \(y=0\), and \(x=b\), where \(b>0\). Let \(\mathscr{R}_{2}\) be the region bounded by \(y=x^{2}\), \(x=0\), and \(y=b^{2}\). (a) Is there a value of \(b\) such that \(\mathscr{R}_{1}\) and \(\mathscr{R}_{2}\) have the same area? (b) Is there a value of \(b\) such that \(\mathscr{R}_{1}\) sweeps out the same volume when rotated about the \(x\)-axis and the \(y\)-axis? (c) Is there a value of \(b\) such that \(\mathscr{R}_{1}\) and \(\mathscr{R}_{2}\) sweep out the same volume when rotated about the \(x\)-axis? (d) Is there a value of \(b\) such that \(\mathscr{R}_{1}\) and \(\mathscr{R}_{2}\) sweep out the same volume when rotated about the \(y\)-axis? Equation Transcription: ?1 ?2 ?1 ?2 ?1 ?1 ?2 ?1 ?2 Text Transcription: ?_1 y = x^2 y = 0 x = b b>0 ?_2 y = x^2 x = 0 y = b^2 b ?_1 ?_2 b ?_1 x y b ?_1 ?_2 x b ?_1 ?_2 y

A solid is generated by rotating about the x-axis the region under the curve y=f(x), where f is a positive function and \(x \geqslant 0\). The volume generated by the part of the curve from x=0 to x=b is \(b^2\) for all b>0. Find the function f.

There is a line through the origin that divides the region bounded by the parabola \(y=x-x^{2}\) and the x-axis into two regions with equal area. What is the slope of that line?

The figure shows a curve C with the property that, for every point P on the middle curve y = 2x2, the areas A and B are equal. Find an equation for C.

A cylindrical glass of radius r and height L is filled with water and then tilted until the water remaining in the glass exactly covers its base. (a) Determine a way to “slice” the water into parallel rectangular cross-sections and then set up a definite integral for the volume of the water in the glass. (b) Determine a way to “slice” the water into parallel cross-sections that are trapezoids and then set up a definite integral for the volume of the water. (c) Find the volume of water in the glass by evaluating one of the integrals in part (a) or part (b). (d) Find the volume of the water in the glass from purely geometric considerations. (e) Suppose the glass is tilted until the water exactly covers half the base. In what direction can you “slice” the water into triangular cross-sections? Rectangular cross-sections? Cross-sections that are segments of circles? Find the volume of water in the glass.

Water in an open bowl evaporates at a rate proportional to the area of the surface of the water. (This means that the rate of decrease of the volume is proportional to the area of the surface.) Show that the depth of the water decreases at a constant rate, regardless of the shape of the bowl.

Archimedes’ Principle states that the buoyant force on an object partially or fully submerged in a fluid is equal to the weight of the fluid that the object displaces. Thus, for an object of density ????0 floating partly submerged in a fluid of density ????????, the buoyant force is given by ???? = ???????????? ????(????)????????, where ???? is the acceleration due to gravity and ????(????) is the area of a typical cross-section of the object (see the figure). The weight of the object is given by (a) Show that the percentage of the volume of the object above the surface of the liquid is (b) The density of ice is 917 kg/m3 and the density of seawater is 1030 kg/m3. What percentage of the volume of an iceberg is above water? (c) An ice cube floats in a glass filled to the brim with water. Does the water overflow when the ice melts? (d) A sphere of radius 0.4 m and having negligible weight is floating in a large freshwater lake. How much work is required to completely submerge the sphere? The density of the water is 1000 kg/m3.

A sphere of radius 1 overlaps a smaller sphere of radius ???? in such a way that their intersection is a circle of radius ????. (In other words, they intersect in a great circle of the small sphere.) Find ???? so that the volume inside the small sphere and outside the large sphere is as large as possible.

A paper drinking cup filled with water has the shape of a cone with height ? and semivertical angle ????. (See the figure.) A ball is placed carefully in the cup, thereby displacing some of the water and making it overflow. What is the radius of the ball that causes the greatest volume of water to spill out of the cup?

A clepsydra, or water clock, is a glass container with a small hole in the bottom through which water can flow. The “clock” is calibrated for measuring time by placing markings on the container corresponding to water levels at equally spaced times. Let ???? = ????(????) be continuous on the interval [0, b] and assume that the container is formed by rotating the graph of ???? about the ????-axis. Let ???? denote the volume of water and h the height of the water level at time ????. (See the figure.) (a) Determine ???? as a function of ?. (b) Show that (c) Suppose that ???? is the area of the hole in the bottom of the container. It follows from Torricelli’s Law that the rate of change of the volume of the water is given by where ???? is a negative constant. Determine a formula for the function ???? such that ?????/???????? is a constant ????. What is the advantage in having ?????/???????? = ?????

A cylindrical container of radius ???? and height ???? is partially filled with a liquid whose volume is ????. If the container is rotated about its axis of symmetry with constant angular speed ????, then the container will induce a rotational motion in the liquid around the same axis. Eventually, the liquid will be rotating at the same angular speed as the container. The surface of the liquid will be convex, as indicated in the figure, because the centrifugal force on the liquid particles increases with the distance from the axis of the container. It can be shown that the surface of the liquid is a paraboloid of revolution generated by rotating the parabola about the ????-axis, where ???? is the acceleration due to gravity. (a) Determine ? as a function of ????. (b) At what angular speed will the surface of the liquid touch the bottom? At what speed will it spill over the top? (c) Suppose the radius of the container is 2 ft, the height is 7 ft, and the container and liquid are rotating at the same constant angular speed. The surface of the liquid is 5 ft below the top of the tank at the central axis and 4 ft below the top of the tank 1 ft out from the central axis. (i) Determine the angular speed of the container and the volume of the fluid. (ii) How far below the top of the tank is the liquid at the wall of the container?

Suppose the graph of a cubic polynomial intersects the parabola ???? = ????2 when ???? = 0, ???? = ????, and ???? = ????, where 0 ? ???? ? ????. If the two regions between the curves have the same area, how is ???? related to ?????

Suppose we are planning to make a taco from a round tortilla with diameter 8 inches by bending the tortilla so that it is shaped as if it is partially wrapped around a circular cylinder. We will fill the tortilla to the edge (but no more) with meat, cheese, and other ingredients. Our problem is to decide how to curve the tortilla in order to maximize the volume of food it can hold. (a) We start by placing a circular cylinder of radius r along a diameter of the tortilla and folding the tortilla around the cylinder. Let x represent the distance from the center of the tortilla to a point P on the diameter (see the figure). Show that the cross-sectional area of the filled taco in the plane through P perpendicular to the axis of the cylinder is \(A(x)=r \sqrt{16-x^2}-\frac{1}{2} r^2 \sin \left(\frac{2}{r} \sqrt{16-x^2}\right)\) and write an expression for the volume of the filled taco. (b) Determine (approximately) the value of r that maximizes the volume of the taco. (Use a graphical approach.)

If the tangent at a point ???? on the curve ???? = ????3 intersects the curve again at ????, let ???? be the area of the region bounded by the curve and the line segment ????????. Let ???? be the area of the region defined in the same way starting with ???? instead of ????. What is the relationship between ???? and ?????

Let ????(????, ????2), a ? 0, be any point on the part of the parabola ???? = ????2 in the first quadrant, and let ? be the region bounded by the parabola and the normal line through ????. (See the figure.) Show that the area of ? is smallest when ???? = . (See also Problem 11 in Problems Plus following Chapter 4.)