53–62. Choose your method Let R be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when R is revolved about the given axis.

y = x²,y=2−x, and x=0, in the first quadrant; about the y-axis

53–62. Choose your method Let R be the region bounded by the following curves. Use the method of your choice to find the volume of the solid generated when R is revolved about the given axis.

y = x,y = 2x+2,x = 2, and x=6; about the y-axis

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

x = y⁴/4 + 1/8y², for 1≤y≤2

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

y = ln (x−√x²−1), for 1 ≤ x ≤ √2(Hint: Integrate with respect to y.)​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​​

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

x = 2y−4, for −3≤y≤4 (Use calculus, but check your work using geometry.)

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

x = 2e^√2y + 1/16e^−√2y, for 0 ≤ y ≤ ln²/√2

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

y = x^3/2 / 3 − x^1/2 on [4, 16]

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]

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

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

y = 3 ln x− x²/24 on [1, 6]

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

y = −8x−3 on [−2, 6] (Use calculus.)

Find the arc length of the line y = 4−3x on [−3, 2] using calculus and verify your answer using geometry.

Find the arc length of the line y = 2x+1 on [1, 5] using calculus and verify your answer using geometry.

3–6. Setting up arc length integrals Write and simplify, but do not evaluate, an integral with respect to x that gives the length of the following curves on the given interval.

y = 2 cos 3x on [−π,π]

Functio​​n 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.

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 = ln x, for 1≤x≤4

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 = x³/3, for −1≤x≤1

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 = cos 2x, for 0 ≤ x ≤ π

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.

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

b. ​If necessary, use technology to evaluate or approximate the integral.

y = cos 2x, for 0 ≤ x ≤ π

Functions from arc length What differentiable functions have an arc length on the interval [a, b] given by the following integrals? Note that the answers are not unique. Give a family of functions that satisfy the conditions.

b. ∫a^b √1+36 cos² 2xdx

Functions from arc length What differentiable functions have an arc length on the interval [a, b] given by the following integrals? Note that the answers are not unique. Give a family of functions that satisfy the conditions.

a. ∫a^b √1+16x⁴ dx

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