9. Graphical Applications of Integrals

69-72. Volumes of solids Find the volume of the following solids.

70. The region bounded by y = 1/[x²(x² + 2)²], y = 0, x = 1, and x = 2 is revolved about the y-axis.

102–105. Volumes The region R is bounded by the curve y = ln(x) and the x-axis on the interval [1, e]. Find the volume of the solid generated when R is revolved in the following ways.

102. About the y-axis

102–105. Volumes The region R is bounded by the curve y = ln(x) and the x-axis on the interval [1, e]. Find the volume of the solid generated when R is revolved in the following ways.

104. About the line y = 1

Area and volume Consider the function f(x) = (9 + x²)^(-1/2) and the region R on the interval [0, 4] (see figure).

b. Find the volume of the solid generated when R is revolved about the x-axis.

119. {Use of Tech} Comparing volumes Let R be the region bounded by y = ln(x), the x-axis, and the line x = a, where a > 1.

b. Find the volume V₂(a) of the solid generated when R is revolved about the y-axis (as a function of a).

120. Equal volumes

a. Let R be the region bounded by the graph of f(x) = x^(-p) and the x-axis, for x ≥ 1. Let V₁ and V₂ be the volumes of the solids generated when R is revolved about the x-axis and the y-axis, respectively, if they exist. For what values of p (if any) is V₁ = V₂?

b. Repeat part (a) on the interval [0, 1].

Surface area of a catenoid When the catenary y = a cosh x/a is revolved about the x-axis, it sweeps out a surface of revolution called a catenoid. Find the area of the surface generated when y = cosh x on [–ln 2, ln 2] is rotated about the x-axis.

43–55. Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions.

What is the volume of the solid whose base is the region in the first quadrant bounded by y = √x,y = 2-x, and the x-axis, and whose cross sections perpendicular to the base and parallel to the y-axis are semicircles?

43–55. Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions.

The region bounded by the curve y = 1+√x, the curve y = 1−√x, and the line x=1 is revolved about the y-axis. Find the volume of the resulting solid by (a) integrating with respect to x and (b) integrating with respect to y. Be sure your answers agree.

43–55. Volumes of solids Choose the general slicing method, the disk/washer method, or the shell method to answer the following questions.

The region bounded by the curves y = sec x and y=2, for 0 ≤ x ≤ π/3, is revolved about the x-axis. What is the volume of the solid that is generated? 

53–62. Choose your method Let R be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when R is revolved about the given axis.

y = x² and y = 2−x²; about the x-axis

Let R be the region bounded by the following curves. Find the volume of the solid generated when R is revolved about the given axis.

y=ln x,y=ln x^2; and y=ln 8; about the y-axis

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. When using the shell method, the axis of the cylindrical shells is parallel to the axis of revolution.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

c. If a region is revolved about the x-axis, then in principle, it is possible to use the disk/washer method and integrate with respect to x or to use the shell method and integrate with respect to y.

53–62. Choose your method Let R be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when R is revolved about the given axis.

y = x−x⁴,y=0; about the x-axis.