9. Graphical Applications of Integrals

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]?

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

c. Write an integral for the volume of the solid.

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]?

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=4−2x, the x-axis, and the y-axis.

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=1−x^3, the x-axis, and 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 line.

x=2−secy,x=2,y=π/3, and y=0; about x=2

The region R is bounded by the graph of f(x)=2x(2−x) and the x-axis. Which is greater, the volume of the solid generated when R is revolved about the line y=2 or the volume of the solid generated when R is revolved about the line y=0? Use integration to justify your answer.

A hemispherical bowl of radius 8 inches is filled to a depth of h inches, where 0≤h≤8 0 ≤ ℎ ≤ 8 . Find the volume of water in the bowl as a function of h. (Check the special cases h=0 and h=8.)

Find the volume of the torus formed when the circle of radius 2 centered at (3, 0) is revolved about the y-axis. Use geometry to evaluate the integral.

A 1.5-mm layer of paint is applied to one side of the following surfaces. Find the approximate volume of paint needed. Assume x and y are measured in meters. 

The spherical zone generated when the curve y=√8x−x^2 on the interval 1≤x≤7 is revolved about the x-axis 

Consider the following curves on the given intervals.  

b. Use a calculator or software to approximate the surface area.

y=tan x , for 0≤x≤π/4; about the x-axis 

9–20. Arc length calculations Find the arc length of the following curves on the given interval.

y = (x²+2)^3/2 / 3 on [0, 1]

Equal integrals Without evaluating integrals, explain the following equalities. (Hint: Draw pictures.)

b. ∫²₀(25−(x²+1)²) dx = 2∫₁⁵ y√y−1 dy

Different axes of revolution Suppose R is the region bounded by y=f(x) and y=g(x) on the interval [a, b], where f(x)≥g(x).

b. How is this formula changed if x0>b?

Explain the steps required to find the length of a curve x = g(y) between y=c and y=d.