The design of boats is based on Archimedes Principle, which states that the buoyant

(a) The region in Figure 8.29 is rotated around the xaxis. Using the strip shown, write an integral giving the volume. (b) Evaluate the integral.

(a) The region in Figure 8.30 is rotated around the xaxis. Using the strip shown, write an integral giving the volume. (b) Evaluate the integral.

(a) The region in Figure 8.31 is rotated around the yaxis. Write an integral giving the volume. (b) Evaluate the integral.

(a) The region in Figure 8.32 is rotated around the yaxis. Using the strip shown, write an integral giving the volume. (b) Evaluate the integral.

In Exercises 514, the region is rotated around the x-axis. Find the volume. 5. Bounded by y = x2, y = 0, x = 0, x = 1.

In Exercises 514, the region is rotated around the x-axis. Find the volume. 5. Bounded by y = x2, y = 0, x = 0, x = 1.

In Exercises 514, the region is rotated around the x-axis. Find the volume. 5. Bounded by y = x2, y = 0, x = 0, x = 1.

In Exercises 514, the region is rotated around the x-axis. Find the volume. 5. Bounded by y = x2, y = 0, x = 0, x = 1.

In Exercises 514, the region is rotated around the x-axis. Find the volume. 5. Bounded by y = x2, y = 0, x = 0, x = 1.

In Exercises 514, the region is rotated around the x-axis. Find the volume. 5. Bounded by y = x2, y = 0, x = 0, x = 1.

In Exercises 514, the region is rotated around the x-axis. Find the volume. 5. Bounded by y = x2, y = 0, x = 0, x = 1.

In Exercises 514, the region is rotated around the x-axis. Find the volume. 5. Bounded by y = x2, y = 0, x = 0, x = 1.

In Exercises 514, the region is rotated around the x-axis. Find the volume. 5. Bounded by y = x2, y = 0, x = 0, x = 1.

In Exercises 514, the region is rotated around the x-axis. Find the volume. 5. Bounded by y = x2, y = 0, x = 0, x = 1.

For Exercises 1520, find the arc length of the graph of the function from x = 0 to x = 2.

For Exercises 1520, find the arc length of the graph of the function from x = 0 to x = 2.

For Exercises 1520, find the arc length of the graph of the function from x = 0 to x = 2.

For Exercises 1520, find the arc length of the graph of the function from x = 0 to x = 2.

For Exercises 1520, find the arc length of the graph of the function from x = 0 to x = 2.

For Exercises 1520, find the arc length of the graph of the function from x = 0 to x = 2.

Find the length of the parametric curves in Exercises 2124.

Find the length of the parametric curves in Exercises 2124.

Find the length of the parametric curves in Exercises 2124.

Find the length of the parametric curves in Exercises 2124.

In Problems 2528 set up, but do not evaluate, an integral that represents the volume obtained when the region in the first quadrant is rotated about the given axis

In Problems 2528 set up, but do not evaluate, an integral that represents the volume obtained when the region in the first quadrant is rotated about the given axis

In Problems 2528 set up, but do not evaluate, an integral that represents the volume obtained when the region in the first quadrant is rotated about the given axis

In Problems 2528 set up, but do not evaluate, an integral that represents the volume obtained when the region in the first quadrant is rotated about the given axis

In Problems 2932, set up definite integral(s) to find the volume obtained when the region between y = x2 and y = 5x is rotated about the given axis. Do not evaluate the integral(s)

In Problems 2932, set up definite integral(s) to find the volume obtained when the region between y = x2 and y = 5x is rotated about the given axis. Do not evaluate the integral(s)

In Problems 2932, set up definite integral(s) to find the volume obtained when the region between y = x2 and y = 5x is rotated about the given axis. Do not evaluate the integral(s)

In Problems 2932, set up definite integral(s) to find the volume obtained when the region between y = x2 and y = 5x is rotated about the given axis. Do not evaluate the integral(s)

Find the length of one arch of y = sin x

Find the perimeter of the region bounded by y = x and y = x

Consider the hyperbola x2 y2 = 1 in Figure 8.33. (a) The shaded region 2 x 3 is rotated around the x-axis. What is the volume generated? (b) What is the arc length with y 0 from x = 2 to x = 3?

Rotating the ellipse x2/a2 + y2/b2 = 1 about the x-axis generates an ellipsoid. Compute its volume.

For Problems 3739, sketch the solid obtained by rotating each region around the indicated axis. Using the sketch, show how to approximate the volume of the solid by a Riemann sum, and hence find the volume.

For Problems 3739, sketch the solid obtained by rotating each region around the indicated axis. Using the sketch, show how to approximate the volume of the solid by a Riemann sum, and hence find the volume.

For Problems 3739, sketch the solid obtained by rotating each region around the indicated axis. Using the sketch, show how to approximate the volume of the solid by a Riemann sum, and hence find the volume.

The solid obtained by rotating the region around the yaxis

The solid obtained by rotating the region about the xaxis.

The solid obtained by rotating the region about the line y = 2.

The solid whose base is the region and whose crosssections perpendicular to the x-axis are squares.

The solid whose base is the region and whose crosssections perpendicular to the x-axis are semicircles

The solid whose base is the region and whose crosssections perpendicular to the y-axis are equilateral triangles

The solid obtained by rotating the region about the xaxis

The solid obtained by rotating the region about the horizontal line y = 3

The solid obtained by rotating the region about the horizontal line y = 7.

The solid whose base is the given region and whose cross-sections perpendicular to the x-axis are squares.

The solid whose base is the given region and whose cross-sections perpendicular to the x-axis are semicircles.

Find a curve y = g(x), such that when the region between the curve and the x-axis for 0 x is revolved around the x-axis, it forms a solid with volume given by , 0 (4 4 cos2 x) dx. [Hint: Use the identity sin2 x = 1 cos2 x.]

A particle starts at the origin and moves along the curve y = 2x3/2/3 in the positive x-direction at a speed of 3 cm/sec, where x, y are in cm. Find the position of the particle at t = 6.

A tree trunk has a circular cross section at every height; its circumference is given in the following table. Estimate the volume of the tree trunk using the trapezoid rule.

Rotate the bell-shaped curve y = ex2/2 shown in Figure 8.34 around the y-axis, forming a hill-shaped solid of revolution. By slicing horizontally, find the volume of this hill.

(a) A pie dish is 9 inches across the top, 7 inches across the bottom, and 3 inches deep. See Figure 8.35. Compute the volume of this dish. (b) Make a rough estimate of the volume in cubic inches of a single cut-up apple, and estimate the number of apples needed to make an apple pie that fills this dish.

A 100 cm long gutter is made of three strips of metal, each 5 cm wide; Figure 8.36 shows a cross-section. (a) Find the volume of water in the gutter when the depth is h cm. (b) What is the maximum value of h? (c) What is the maximum volume of water that the gutter can hold? (d) If the gutter is filled with half the maximum volume of water, is the depth larger or smaller than half of the answer to part (b)? Explain how you can answer without any calculation. (e) Find the depth of the water when the gutter contains half the maximum possible volume

The design of boats is based on Archimedes Principle, which states that the buoyant force on an object in water is equal to the weight of the water displaced. Suppose you want to build a sailboat whose hull is parabolic with cross section y = ax2, where a is a constant. Your boat will have length L and its maximum draft (the maximum vertical depth of any point of the boat beneath the water line) will be H. See Figure 8.37. Every cubic meter of water weighs 10,000 newtons. What is the maximum possible weight for your boat and cargo?

The circumference of a tree at different heights above the ground is given in the table below. Assume that all horizontal cross-sections of the tree are circles. Estimate the volume of the tree.

A bowl has the shape of the graph of y = x4 between the points (1, 1) and (1, 1) rotated about the y-axis. When the bowl contains water to a depth of h units, it flows out through a hole in the bottom at a rate (volume/time) proportional to h, with constant of proportionality 6. (a) Show that the water level falls at a constant rate. (b) Find how long it takes to empty the bowl if it is originally full to the brim.

The hull of a boat has widths given by the following table. Reading across a row of the table gives widths at points 0, 10, . . . , 60 feet from the front to the back at a certain level below waterline. Reading down a column of the table gives widths at levels 0, 2, 4, 6, 8 feet below waterline at a certain distance from the front. Use the trapezoidal rule to estimate the volume of the hull below waterline

(a) Write an integral which represents the circumference of a circle of radius r. (b) Evaluate the integral, and show that you get the answer you expect

Compute the perimeter of the region used for the base of the solids in Problems 4650.

Write an integral that represents the arc length of the portion of the graph of f(x) = x(x 4) that lies above the x-axis. Do not evaluate the integral

Find a curve y = f(x) whose arc length from x = 1 to x = 4 is given by , 4 1 1 + x dx

Write a simplified expression that represents the arc length of the concave-down portion of the graph of f(x) = ex2 . Do not evaluate your answer.

Write an expression that represents the arc length of the concave-down portion of the graph of f(x) = x4 8x3 + 18x2 + 3x + 7. Do not simplify or evaluate the answer.

With x and b in meters, a chain hangs in the shape of the catenary cosh x = 1 2 (ex + ex) for b x b. If the chain is 10 meters long, how far apart are its ends?

There are very few elementary functions y = f(x) for which arc length can be computed in elementary terms using the formula , b a 1 + dy dx2 dx. You have seen some such functions f in Problems 18, 19, and 67, namely, f(x) = x3, f(x) = 4 x2, and f(x) = 1 2 (ex + ex). Try to find some other function that works, that is, a function whose arc length you can find using this formula and antidifferentiation

After doing Problem 68, you may wonder what sort of functions can represent arc length. If g(0) = 0 and g is differentiable and increasing, then can g(x), x 0, represent arc length? That is, can we find a function f(t) such that , x 0 1+(f(t))2 dt = g(x)? (a) Show that f(x) = - x 0 (g(t))2 1 dt works as long as g (x) 1. In other words, show that the arc length of the graph of f from 0 to x is g(x). (b) Show that if g (x) < 1 for some x, then g(x) cannot represent the arc length of the graph of any function. (c) Find a function f whose arc length from 0 to x is 2x.

Consider the graph of the equation |x| k + |y| k = 1, k constant. For k an even integer, the absolute values are unnecessary. For example, for k = 2, we see the equation gives the circle x2 + y2 = 1. (a) Sketch the graph of the equation for k = 1, 2, 4. (b) Find the arc length of the three graphs in part (a). [Note: k = 4 may require a computer.]

In Problems 7173, explain what is wrong with the statement

In Problems 7173, explain what is wrong with the statement

In Problems 7173, explain what is wrong with the statement

A region in the plane which gives the same volume whether rotated about the x-axis or the y-axis.

A region where the solid obtained by rotating the region around the x-axis has greater volume than the solid obtained by revolving the region around the y-axis.

Two different curves from (0, 0) to (10, 0) that have the same arc length

A function f(x) whose graph passes through the points (0, 0) and (1, 1) and whose arc length between x = 0 and x = 1 is greater than 2

Of two solids of revolution, the one with the greater volume is obtained by revolving the region in the plane with the greater area

f f is differentiable on the interval [0, 10], then the arc length of the graph of f on the interval [0, 1] is less than the arc length of the graph of f on the interval [1, 10].

If f is concave up for all x and f (0) = 3/4, then the arc length of the graph of f on the interval [0, 4] is at least 5

If f is concave down for all x and f (0) = 3/4, then the arc length of the graph of f on the interval [0, 4] is at most 5