9. Graphical Applications of Integrals

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

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^2,y=2−x, and y=0; about the y-axis

Function defined as an integral Write the integral that gives the length of the curve y = f(x) = ∫₀^x sin t dt on the interval [0,π]

Lengths of symmetric curves Suppose a curve is described by y=f(x) on the interval [−b, b], where f′ is continuous on [−b, b]. Show that if f is odd or f is even, then the length of the curve y=f(x) from x=−b to x=b is twice the length of the curve from x=0 to x=b. Use a geometric argument and prove it using integration.

58–61. Arc length Find the length of the following curves.

y = 2x+4 on [−2,2] (Use calculus.)

Volume of a sphere Let R be the region bounded by the upper half of the circle x²+y² = r² and the x-axis. A sphere of radius r is obtained by revolving R about the x-axis.

a. Use the shell method to verify that the volume of a sphere of radius r is 4/3 πr³.

Volume of a sphere Let R be the region bounded by the upper half of the circle x²+y² = r² and the x-axis. A sphere of radius r is obtained by revolving R about the x-axis.

b. Repeat part (a) using the disk method.

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].

b. Find the volume of the solid generated when R is revolved about the y-axis.

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.

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

b. Use the washer method to write an integral for the volume of the torus.

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.

Surface area of a cone Find the surface area of a cone (excluding the base) with radius 4 and height 8 using integration and a surface area integral.

21–30. {Use of Tech} Arc length by calculator

a. Write and simplify the integral that gives the arc length of the following curves on the given interval. 

y = 1/x, for 1 ≤ x ≤ 10

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

Arc length may be negative if f(x) < 0 on part of the interval in question.

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.)