9. Graphical Applications of Integrals

65. Volume Find the volume of the solid generated when the region bounded by y = sin²(x) * cos^(3/2)(x) and the x-axis on the interval [0, π/2] is revolved about the x-axis.

87. Surface area Find the area of the surface generated when the curve f(x) = sin x on [0, π/2] is revolved about the x-axis.

Find the area of the surface generated when the given curve is revolved about the given axis.

y=3x+4, for 0≤x≤6; about the x-axis

Find the area of the surface generated when the given curve is revolved about the given axis.

y=8√x, for 9≤x≤20; about the x-axis

Find the area of the surface generated when the given curve is revolved about the given axis.

y=(3x)^1/3 , for 0≤x≤8/3; about the y-axis

Find the area of the surface generated when the given curve is revolved about the given axis.

y=√1−x^2, for −1/2≤x≤1/2; about the x-axis

Find the area of the surface generated when the given curve is revolved about the given axis.

y=4x−1, for 1≤x≤4; about the y-axis (Hint: Integrate with respect to y.)

Find the area of the surface generated when the given curve is revolved about the given axis.

y=1/4(e^2x+e^−2x), for −2≤x≤2; about the x-axis

Find the area of the surface generated when the given curve is revolved about the given axis.

x=√12y−y^2, for 2≤y≤10; about the y-axis

What is the area of the curved surface of a right circular cone of radius 3 and height 4?

Assume f is a nonnegative function with a continuous first derivative on [a, b]. The curve y=f(x) on [a, b] is revolved about the x-axis. Explain how to find the area of the surface that is generated.

A surface is generated by revolving the line f(x)=2−x, for 0≤x≤2, about the x-axis. Find the area of the resulting surface in the following ways.

a. Using calculus

Let R be the region in the first quadrant bounded above by the curve y=2−x² and bounded below by the line y=x. Suppose the shell method is used to determine the volume of the solid generated by revolving R about the y-axis.

a. What is the radius of a cylindrical shell at a point x in [0, 2]?

Let R be the region in the first quadrant bounded above by the curve y=2−x² and bounded below by the line y=x. Suppose the shell method is used to determine the volume of the solid generated by revolving R about the y-axis.

b. What is the height of a cylindrical shell at a point x in [0, 2]?

Let R be the region in the first quadrant bounded above by the curve y=2−x² and bounded below by the line y=x. Suppose the shell method is used to determine the volume of the solid generated by revolving R about the y-axis.

c. Write an integral for the volume of the solid using the shell method.