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

Region R is revolved about the line x=4 to form a solid of revolution.

a. What is the radius of a cross section of the solid at a point y in [1, 3]?

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.

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.

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.

A right circular cylinder with height R and radius R has a volume of VC=πR^3 (height = radius).

a. Find the volume of the cone that is inscribed in the cylinder with the same base as the cylinder and height R. Express the volume in terms of VC.

74. Volume of a Solid

Consider the region R bounded by:

The graph of f(x) = 1/(x + 2)

The x-axis on the interval [0,3].

Find the volume of the solid formed when R is revolved about the y-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

39–44. Shell method about other lines Let R be the region bounded by y = x²,x=1, and y=0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines.

y = 2

9-34. Shell method Let R be the region bounded by the following curves. Use the shell method to find the volume of the solid generated when R is revolved about indicated axis. 

{Use of Tech} y = √50 -2x², in the first quadrant; about the x-axis

39–44. Shell method about other lines Let R be the region bounded by y = x²,x=1, and y=0. Use the shell method to find the volume of the solid generated when R is revolved about the following lines.

y = -2

27–33. Multiple regions The regions R₁,R₂, and R₃ (see figure) are formed by the graphs of y = 2√x,y = 3−x,and x=3.

Find the volume of the solid obtained by revolving region R₂ about the y-axis.

Volumes

Find the volume of the solid generated by revolving the region bounded by the parabola y² = 4x and the line y = x about

d. the line y = 4

A torus (doughnut) A torus is formed when a circle of radius 2 centered at (3, 0) is revolved about the y-axis.

a. Use the shell method to write an integral for the volume of the torus.

Why is the disk method a special case of the general slicing method?