Chapter 6

Find the average value of the function on the given interval. \( f(x) = 3x^2 +8x \) , \( [-1, 2] \)

A 360-lb gorilla climbs a tree to a height of 20 ft. Find the work done if the gorilla reaches that height in (a) 10 seconds (b) 5 seconds

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. \( y = x + 1 \) , \( y = 0 \) , \( x = 0 \) , \( x = 2 \) ; about the x-axis

Find the average value of the function on the given interval. \( f(x) = \sqrt{x} \) , \( [0, 4] \)

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. \( y = \frac{1}{x} \) , \( y = 0 \) , \( x = 1 \) , \( x = 4 \) ; about the x-axis

Find the average value of the function on the given interval. \( g(x) = 3 \cos x \) , \( [\frac{-\pi}{2}, \frac{\pi}{2}] \)

A variable force of \( 5x^{-2} \) pounds moves an object along a straight line when it is \( x \) feet from the origin. Calculate the work done in moving the object from \( x = 1 ft \) to \( = 10 ft \).

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis. \( y = \sqrt[3]{x} \) , \( y = 0 \) , \( x = 1 \)

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. \( y = \sqrt{x - 1} \) , \( y = 0 \) , \( x = 5 \) ; about the x-axis

Find the average value of the function on the given interval. \( g(t) = \dfrac{t}{\sqrt{3 + t^2}} \) , \( [1, 3] \)

When a particle is located a distance \( x \) meters from the origin, a force of \( \cos (\frac{\pi x}{3}) \) newtons acts on it. How much work is done in moving the particle from \( x = 1 \) to \( x = 2 \)? Interpret your answer by considering the work done from \( x = 1 \) to \( x = 1.5 \) and from \( x = 1.5 \) to \( x = 2 \).

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis. \( y = x^3 \) , \( y = 0 \) , \( x = 1 \) , \( x = 2 \)

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. \( y = e^x \) , \( y = 0 \) , \( x = -1 \) , \( x = 1 \) ; about the x-axis

Find the average value of the function on the given interval. \( f(t) = e^{\sin t} \cos t \) , \( [0, \frac{\pi}{2}] \)

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis. \( y = e^{-x^2} \) , \( y = 0 \) , \( x = 0 \) , \( x = 1 \)

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. \( x = 2 \sqrt{y} \) , \( x = 0 \) , \( y = 9 \) ; about the y-axis

Sketch the region enclosed by the given curves. Decide whether to integrate with respect to \( x \) and \( y \). Draw a typical approximating rectangle and label its height and width. Then find the area of the region. \( y = e^x \) , \( y = x^2 - 1 \) , \( x = -1 \) , \( x = 1 \)

Find the average value of the function on the given interval. \( f(x) = \dfrac{x^2}{(x^3 + 3)^2} \) , \( [-1, 1] \)

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis. \( y = 4x - x^2 \) , \( y = x \)

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. \( 2x = y^2 \) , \( x = 0 \) , \( y = 4 \) ; about the y-axis

Sketch the region enclosed by the given curves. Decide whether to integrate with respect to \( x \) and \( y \). Draw a typical approximating rectangle and label its height and width. Then find the area of the region. \( y = \sin x \) , \( y = x \) , \( x = \frac{\pi}{2} \) , \( x = \pi \)

Find the average value of the function on the given interval. \( h(x) = \cos^4 x \sin x \) , \( [0, \pi] \)

A force of 10 lb is required to hold a spring stretched 4 in. beyond its natural length. How much work is done in stretching it from its natural length to 6 in. beyond its natural length?

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis. \( y = x^2 \) , \( y = 6x - 2x^2 \)

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. \( y = x^3 \) , \( y = x \) , \( x \ge 0 \) ; about the x-axis

Sketch the region enclosed by the given curves. Decide whether to integrate with respect to \( x \) and \( y \). Draw a typical approximating rectangle and label its height and width. Then find the area of the region. \( y = (x - 2)^2 \) , \( y = x \)

Find the average value of the function on the given interval. \( h(u) = \dfrac{(\ln u)}{u} \) , \( [1, 5] \)

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. \( y = 6 - x^2 \) , \( y = 2 \) ; about the x-axis

Sketch the region enclosed by the given curves. Decide whether to integrate with respect to \( x \) and \( y \). Draw a typical approximating rectangle and label its height and width. Then find the area of the region. \( y = x^2 - 4x \) , \( y = 2x \)

(a) Find the average value of \( f \) on the given interval. (b) Find \( c \) such that \( f_{ave} = f(c) \). (c) Sketch the graph of \( f \) and a rectangle whose area is the same as the area under the graph of \( f \). \( f(x) = (x - 3)^2 \) , \( [2, 5] \)

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. \( xy = 1 \) , \( x = 0 \) , \( y = 1 \) , \( y = 3 \)

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. \( y^2 = x \) , \( x = 2y \) ; about the y-axis

(a) Find the average value of \( f \) on the given interval. (b) Find \( c \) such that \( f_{ave} = f(c) \). (c) Sketch the graph of \( f \) and a rectangle whose area is the same as the area under the graph of \( f \). \( f(x) = \dfrac{1}{x} \) , \( [1, 3] \)

If the work required to stretch a spring 1 ft beyond its natural length is 12 ft-lb, how much work is needed to stretch it 9 in. beyond its natural length?

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. \( y = \sqrt{x} \) , \( x = 0 \) , \( y = 2 \)

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. \( x = -2 - y^2 \) , \( x = y^4 \) ; about the y-axis

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. \( y = x^{\frac{3}{2}} \) , \( y = 8 \) , \( x = 0 \)

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. \( y = x^2 \) , \( x = y^2 \) ; about \( y = 1 \)

Sketch the region enclosed by the given curves. Decide whether to integrate with respect to \( x \) and \( y \). Draw a typical approximating rectangle and label its height and width. Then find the area of the region. \( x = 1 - y^2 \) , \( x = y^2 - 1 \)

(a) Find the average value of \( f \) on the given interval. (b) Find \( c \) such that \( f_{ave} = f(c) \). (c) Sketch the graph of \( f \) and a rectangle whose area is the same as the area under the graph of \( f \). \( f(x) = 2xe^{-x^2} \) , \( [0, 2] \)

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. \( y = x^3 \) , \( y = 1 \) , \( x = 2 \) ; about \( y = -3 \)

Sketch the region enclosed by the given curves. Decide whether to integrate with respect to \( x \) and \( y \). Draw a typical approximating rectangle and label its height and width. Then find the area of the region. \( 4x + y^2 = 12 \) , \( x = y \)

If \( f \) is continuous and \( \displaystyle \int_{1}^3 f(x) dx = 8 \), show that \( f \) takes on the value 4 at least once on the interval \( [1, 3] \).

Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A heavy rope, 50 ft long, weighs 0.5 lb/ft and hangs over the edge of a building 120 ft high. (a) How much work is done in pulling the rope to the top of the building? (b) How much work is done in pulling half the rope to the top of the building?

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. \( x = 1 + (y - 2)^2 \) , \( x = 2 \)

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. \( y = 1 + \sec x \) , \( y = 3 \) ; about \( y = 1 \)

Sketch the region enclosed by the given curves and find its area. \( y = 12 - x^2 \) , \( y = x^2 - 6 \)

Find the numbers \( b \) such that the average value of \( f(x) = 2 + 6x - 3x^2 \) on the interval \( [0, b] \) is equal to 3.

Use the method of cylindrical shells to find the volume of the solid obtained by rotating the region bounded by the given curves about the x-axis. \( x + y = 4 \) , \( x = y^2 - 4y + 4 \)

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. \( y = \sin x \) , \( y = \cos x \) , \( 0 \le x \le \frac{\pi}{4} \) ; about \( y = -1 \)