9. Graphical Applications of Integrals

For the following regions R, determine which is greater—the volume of the solid generated when R is revolved about the x-axis or about the y-axis.

R is bounded by y=x^2 and y=√8x.

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.

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 graph of y = 4−x² and the x-axis on the interval [−2,2] is revolved about the line x = −2. What is the volume of the solid that is generated?

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

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

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.

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

y=x^3/2−x^1/2 / 3, for 1≤x≤2; about the x-axis 

Region R is revolved about the line y=1 to form a solid of revolution.

a. What is the radius of a cross section of the solid at a point x in [0, 4]?

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.

Consider the region R in the first quadrant bounded by y=x^1/n and y=x^n, where n>1 is a positive number.

a. Find the volume V(n) of the solid generated when R is revolved about the x-axis. Express your answer in terms of n.

Volumes without calculus Solve the following problems with and without calculus. A good picture helps.

b. A cube is inscribed in a right circular cone with a radius of 1 and a height of 3. What is the volume of the cube?

Use the general slicing method to find the volume of the following solids.

The solid whose base is the triangle with vertices (0, 0), (2, 0), and (0, 2), and whose cross sections perpendicular to the base and parallel to the y-axis are semicircles

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=2x,y=0 , and x=3; about the x-axis (Verify that your answer agrees with the volume formula for a cone.)

Surface area and volume Let f(x) = 1/3 x³ and let R be the region bounded by the graph of f and the x-axis on the interval [0, 2].

c. Find the volume of the solid generated when R is revolved 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.

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?