9. Graphical Applications of Integrals

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=sin xon [0,π] and y=0 ; about the x-axis (Hint: Recall that sin^2 x=1 − cos2x / 2.

64–68. Shell method Use the shell method to find the volume of the following solids.

A right circular cone of radius 3 and height 8

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

b. If a region is revolved about the y-axis, then the shell method must be used.

65-76. Volumes Find the volume of the described solid of revolution or state that it does not exist.

66. The region bounded by f(x) = (x^2 + 1)^(-1/2) and the x-axis on the interval [2, ∞) is revolved about the x-axis.

65-76. Volumes Find the volume of the described solid of revolution or state that it does not exist.

69. The region bounded by f(x) = 1/√(x ln x) and the x-axis on the interval [e, ∞) is revolved about the x-axis.

65-76. Volumes Find the volume of the described solid of revolution or state that it does not exist.

72. The region bounded by f(x) = (x + 1)^(-3/2) and the x-axis on the interval (-1, 1] is revolved about the line y = -1.

65-76. Volumes Find the volume of the described solid of revolution or state that it does not exist.

75. The region bounded by f(x) = (4 - x)^(-1/3) and the x-axis on the interval [0, 4) is revolved about the y-axis.

64–68. Shell method Use the shell method to find the volume of the following solids.

The solid formed when a hole of radius 3 is drilled symmetrically along the axis of a right circular cone of radius 6 and height 9

42-47. Volumes of Solids Find the volume of the solid generated when the given region is revolved as described.

42. The region bounded by f(x) = ln(x), y = 1, and the coordinate axes is revolved about the x-axis.

69. Comparing volumes Let R be the region bounded by y = sin x and the x-axis on the interval [0, π]. Which is greater, the volume when R is revolved about the x-axis, or the volume when R is revolved about the y-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=√sin x,y=1, and x=0; 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=|x| and y=2−x^2; about the x-axis 

Two methods The region R in the first quadrant bounded by the parabola y = 4-x² and coordinate axes is revolved about the y-axis to produce a dome-shaped solid. Find the volume of the solid in the following ways:

a. Apply the disk method and integrate with respect to y.

Two methods The region R in the first quadrant bounded by the parabola y = 4-x² and coordinate axes is revolved about the y-axis to produce a dome-shaped solid. Find the volume of the solid in the following ways:

b. Apply the shell method and integrate with respect to x.

Area and volume The region R is bounded by the curves x = y²+2,y=x−4, and y=0 (see figure).

b. Write a single integral that gives the volume of the solid generated when R is revolved about the x-axis.