Chapter 6

Sketch the region enclosed by the given curves and find its area. \( y = x^2 \) , \( y = 4x - x^2 \)

Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A cable that weighs 2 lb/ft is used to lift 800 lb of coal up a mine shaft 500 ft deep. Find the work done.

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis. \( y = x^3 \) , \( y = 8 \) , \( x = 0 \) ; about \( x = 3 \)

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. \( y = x^3 \) , \( y = 0 \) , \( x = 1 \) ; about \( x = 2 \)

Sketch the region enclosed by the given curves and find its area. \( y = \sec^2 x \) , \( y = 8 \cos x \) , \( \frac{-\pi}{3} \le x \le \frac{\pi}{3} \)

Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A chain lying on the ground is 10 m long and its mass is 80 kg. How much work is required to raise one end of the chain to a height of 6 m?

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis. \( y = 4 - 2x \) , \( y = 0 \) , \( x = 0 \) ; about \( x = -1 \)

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. \( xy = 1 \) , \( y = 0 \) , \( x = 1 \) , \( x = 2 \) ; about \( x = -1 \)

Sketch the region enclosed by the given curves and find its area. \( y = \cos x \) , \( y = 2 - \cos x \) , \( 0 \le x \le 2\pi \)

In a certain city the temperature (in \( ^\circ F \)) \( t \) hours after 9 am was modeled by the function $$ T(t) = 50 + 14 \sin \frac{\pi t}{12} $$ Find the average temperature during the period from 9 am to 9 pm.

Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A leaky 10-kg bucket is lifted from the ground to a height of 12 m at a constant speed with a rope that weighs 0.8 kg/m. Initially the bucket contains 36 kg of water, but the water leaks at a constant rate and finishes draining just as the bucket reaches the 12-m level. How much work is done?

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis. \( y = 4x - x^2 \) , \( y = 3 \) ; about \( x = 1 \)

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. \( x = y^2 \) , \( x = 1 - y^2 \) ; about \( x = 3 \)

Sketch the region enclosed by the given curves and find its area. \( x = 2y^2 \) , \( x = 4 + y^2 \)

The velocity \( v \) of blood that flows in a blood vessel with radius \( R \) and length \( l \) at a distance \( r \) from the central axis is $$ v(r) = \frac{P}{4\eta l} (R^2 - r^2) $$ where \( P \) is the pressure difference between the ends of the vessel and \( \eta \) is the viscosity of the blood (see Example 3.7.7). Find the average velocity (with respect to \( r \)) over the interval \( 0 \le r \le R \). Compare the average velocity with the maximum velocity.

Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A bucket that weighs 4 lb and a rope of negligible weight are used to draw water from a well that is 80 ft deep. The bucket is filled with 40 lb of water and is pulled up at a rate of 2 ft/s, but water leaks out of a hole in the bucket at a rate of 0.2 lb/s. Find the work done in pulling the bucket to the top of the well.

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified axis. \( y = \sqrt{x} \) , \( x = 2y \) ; about \( x = 5 \)

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. $$ y=x, y=0, x=2, x=4 ; \quad \text { about } x=1 $$

Sketch the region enclosed by the given curves and find its area. \( y = \sqrt{x - 1} \) , \( x - y = 1 \)

The linear density in a rod 8 m long is \( \frac{12}{\sqrt{x + 1}} \) kg/m, where \( x \) is measured in meters from one end of the rod. Find the average density of the rod.

Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A 10-ft chain weighs 25 lb and hangs from a ceiling. Find the work done in lifting the lower end of the chain to the ceiling so that it's level with the upper end.

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

Sketch the region enclosed by the given curves and find its area. \( y = \cos \pi x \) , \( y = 4x^2 - 1 \)

Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. A circular swimming pool has a diameter of 24 ft, the sides are 5 ft high, and the depth of the water is 4 ft. How much work is required to pump all of the water out over the side? (Use the fact that water weighs \( 62.5 lb/ft^3 \).)

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

Show how to approximate the required work by a Riemann sum. Then express the work as an integral and evaluate it. An aquarium 2 m long, 1 m wide, and 1 m deep is full of water. Find the work needed to pump half of the water out of the aquarium. (Use the fact that the density of water is \( 1000 kg/m^3 \).)

(a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis. (b) Use your calculator to evaluate the integral correct to five decimal places. \( y = xe^{-x} \) , \( y = 0 \) , \( x = 2 \) ; about the y-axis

Sketch the region enclosed by the given curves and find its area. \( y = \tan x \) , \( y = 2 \sin x \) , \( \frac{-\pi}{3} \le x \le \frac{\pi}{3} \)

If a freely falling body starts from rest, then its displacement is given by \( s = \frac{1}{2} gt^2 \). Let the velocity after a time \( T \) be \( v_T \). Show that if we compute the average of the velocities with respect to \( t \) we get \( v_{ave} = \frac{1}{2} v_T \), but if we compute the average of the velocities with respect to s we get \( v_{ave} = \frac{2}{3} v_T \).

Sketch the region enclosed by the given curves and find its area. \( y = x ^3 \) , \( y = x \)

(a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis. (b) Use your calculator to evaluate the integral correct to five decimal places. \( y = \cos^4 x \) , \( y = -\cos^4 x \) , \( \frac{-\pi}{2} \le x \le \frac{\pi}{2} \) ; about \( x = \pi \)

Sketch the region enclosed by the given curves and find its area. \( y = \sqrt[3]{2x} \) , \( y = \frac{1}{8}x^2 \) , \( 0 \le x \le 6 \)

Sketch the region enclosed by the given curves and find its area. \( y = \cos x \) , \( y = 1 - \cos x \) , \( 0 \le x \le \pi \)

(a) Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curve about the specified axis. (b) Use your calculator to evaluate the integral correct to five decimal places. \( x = \sqrt{\sin y} \) , \( 0 \le y \le \pi \) , \( x = 0 \) ; about \( y = 4 \)

Sketch the region enclosed by the given curves and find its area. \( y = x^4 \) , \( y = 2 - \mid x \mid \)

If \( f_{ave} [a, b] \) denotes the average value of \( f \) on the interval \( [a, b] \) and \( a < c < b \), show that $$ f_{ave} [a, b] = \frac{c - a}{b - a} f_{ave} [a, c] + \frac{b - c}{b - a} f_{ave} [c, b] $$

Sketch the region enclosed by the given curves and find its area. \( y = \sinh x \) , \( y = e^{-x} \) , \( x = 0 \) , \( x = 2 \)

Use the Midpoint Rule with \( n = 5 \) to estimate the volume obtained by rotating about the y-axis the region under the curve \( y = \sqrt{1 + x^3} \) , \( 0 \le x \le 1 \).

Sketch the region enclosed by the given curves and find its area. \( y = \frac{1}{x} \) , \( y = x \) , \( y = \frac{1}{4}x \) , \( x > 0 \)

Sketch the region enclosed by the given curves and find its area. \( y = \frac{1}{4}x^2 \) , \( y = 2x^2 \) , \( x + y = 3 \) , \( x \ge 0 \)

When gas expands in a cylinder with radius \( r \), the pressure at any given time is a function of the volume: \( P = P(V) \). The force exerted by the gas on the piston (see the figure) is the product of the pressure and the area: \( F = \pi r^2 P \). Show that the work done by the gas when the volume expands from volume \( V_1 \) to volume \( V_2 \) is $$ W = \int_{V_1}^{V_2} P dV $$

Sketch the region enclosed by the given curves and find its area. \( y = \frac{x}{\sqrt{1 + x^2}} \) , \( y = \frac{x}{\sqrt{9 - x^2}} \) , \( x \ge 0 \)

The kinetic energy \( KE \) of an object of mass m moving with velocity \( v \) is defined as \( KE = \frac{1}{2} mv^2 \). If a force \( f(x) \) acts on the object, moving it along the x-axis from \( x_1 \) to \( x_2 \), the Work-Energy Theorem states that the net work done is equal to the change in kinetic energy: \( \frac{1}{2} mv^2_2 - \frac{1}{2} mv^2_1\) , where \( v_1 \) is the velocity at \( x_1 \) and \( v_2 \) is the velocity at \( x_2 \). (a) Let \( x = s(t) \) be the position function of the object at time \( t \) and \( v(t) \), \( a(t) \) the velocity and acceleration functions. Prove the Work- Energy Theorem by first using the Substitution Rule for Definite Integrals (5.5.6) to show that $$ W = \int_{x_1}^{x_2} f(x) dx = \int_{t_1}^{t_2} f(s(t)) v(t) dt $$ The use Newton's Second Law of Motion (force = mass \( \times \) acceleration) and the substitution \( u = v(t) \) to evaluate the integral. (b) How much work (in ft-lb) is required to hurl a 12-lb bowling ball at 20 mi/h? (Note: Divide the weight in pounds by \( 32 ft/s^2 \), the acceleration due to gravity, to find the mass, measure in slugs.)

Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places. \( y = e^{-x^2} \) , \( y = 0 \) , \( x = -1 \) , \( x = 1 \) (a) About the x-axis (b) About \( y = -1 \)

Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places. \( y = 0 \) , \( y = \cos^2 x \) , \( \frac{-\pi}{2} \le x \le \frac{\pi}{2} \) (a) About the x-axis (b) About \( y = 1 \)

(a) Newton's Law of Gravitation states that two bodies with masses \( m_1 \) and \( m_2 \) attract each other with a force $$ F = G \frac{m_1 m_2}{r^2} $$ where \( r \) is the distance between the bodies and \( G \) is the gravitational constant. If one of the bodies is fixed, find the work needed to move the other from \( r = a \) to \( r = b \). (b) Compute the work required to launch a 1000-kg satellite vertically to a height of 1000 km. You may assume that the earth's mass is \( 5.98 \times 10^{24} kg \) and is concentrated at its center. Take the radius of the earth to be \( 6.37 \times 10^6 m \) and \( G = 6.67 \times 10^{-11} N \cdot m^2/kg^2 \).

Use calculus to find the area of the triangle with the given vertices. \( (0 , 0) \) , \( (3 , 1) \) , \( (1 , 2) \)

The Great Pyramid of King Khufu was built of limestone in Egypt over a 20-year time period from 2580 bc to 2560 BC. Its base is a square with side length 756 ft and its height when built was 481 ft. (It was the tallest man-made structure in the world for more than 3800 years.) The density of the limestone is about \( 150 lb/ft^3 \). (a) Estimate the total work done in building the pyramid. (b) If each laborer worked 10 hours a day for 20 years, for 340 days a year, and did 200 ft-lb/h of work in lifting the limestone blocks into place, about how many laborers were needed to construct the pyramid?

Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places. \( y =x^2 \) , \( x^2 + y^2 = 1 \) , \( y \ge 0 \) (a) About the x-axis (b) About the y-axis

Use calculus to find the area of the triangle with the given vertices. \( (2 , 0) \) , \( (0 , 2) \) , \( (-1 , 1) \)