Chapter 5

In Exercises \(1-4\), set up the definite integral that gives the area of the region. $$ \begin{array}{l} f(x)=x^{2}-6 x \\ g(x)=0 \end{array} $$

Solve the differential equation. $$ \frac{d y}{d x}=x+2 $$

In Exercises 1 and 2 , find the distance between the points using (a) the Distance Formula and (b) integration. $$ (0,0), \quad(5,12) $$

Constant Force In Exercises 1 and 2 , determine the work done by the constant force. A 100 -pound bag of sugar is lifted 10 feet.

Solve the differential equation. $$ \frac{d y}{d x}=4-y $$

In Exercises 1 and 2 , find the distance between the points using (a) the Distance Formula and (b) integration. $$ (1,2), \quad(7,10) $$

Constant Force In Exercises 1 and 2 , determine the work done by the constant force. An electric hoist lifts a 2800 -pound car 4 feet.

Set up the definite integral that gives the area of the region. $$ \begin{array}{l} f(x)=3\left(x^{3}-x\right) \\ g(x)=0 \end{array} $$

Solve the differential equation. $$ y^{\prime}=\frac{5 x}{y} $$

In Exercises \(3-14,\) find the arc length of the graph of the function over the indicated interval. $$ y=\frac{2}{3} x^{3 / 2}+1, \quad[0,1] $$

Hooke's Law In Exercises 3-10, use Hooke's Law to determine the variable force in the spring problem. A force of 5 pounds compresses a 15 -inch spring a total of 4 inches. How much work is done in compressing the spring 7 inches?

Set up the definite integral that gives the area of the region. $$ \begin{array}{l} f(x)=(x-1)^{3} \\ g(x)=x-1 \end{array} $$

Solve the differential equation. $$ y^{\prime}=\frac{\sqrt{x}}{3 v} $$

In Exercises \(3-14,\) find the arc length of the graph of the function over the indicated interval. $$ y=2 x^{3 / 2}+3, \quad[0,9] $$

In Exercises \(5-8,\) the integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. $$ \int_{0}^{4}\left[(x+1)-\frac{x}{2}\right] d x $$

Solve the differential equation. $$ y^{\prime}=\sqrt{x} y $$

In Exercises \(3-14,\) find the arc length of the graph of the function over the indicated interval. $$ y=\frac{3}{2} x^{2 / 3}, \quad[1,8] $$

Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the \(y\) -axis. $$ y=x^{2}, \quad y=0, \quad x=2 $$

The integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. $$ \int_{2}^{3}\left[\left(\frac{x^{3}}{3}-x\right)-\frac{x}{3}\right] d x $$

Solve the differential equation. $$ y^{\prime}=x(1+y) $$

In Exercises \(3-14,\) find the arc length of the graph of the function over the indicated interval. $$ y=\frac{x^{4}}{8}+\frac{1}{4 x^{2}}, \quad[1,2] $$

Hooke's Law In Exercises 3-10, use Hooke's Law to determine the variable force in the spring problem. A force of 800 newtons stretches a spring 70 centimeters on a mechanical device for driving fence posts. Find the work done in stretching the spring the required 70 centimeters.

Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the \(y\) -axis. $$ y=\frac{1}{2} x^{2}, \quad y=0, \quad x=6 $$

The integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. $$ \int_{-\pi / 3}^{\pi / 3}(2-\sec x) d x $$

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines. \(y=\sqrt{x}, \quad y=0, \quad x=4\) (a) the \(x\) -axis (b) the \(y\) -axis (c) the line \(x=4\) (d) the line \(x=6\)

Solve the differential equation. $$ \left(1+x^{2}\right) y^{2}-2 x y=0 $$

In Exercises \(3-14,\) find the arc length of the graph of the function over the indicated interval. $$ y=\frac{x^{5}}{10}+\frac{1}{6 x^{3}}, \quad[1,2] $$

Hooke's Law In Exercises 3-10, use Hooke's Law to determine the variable force in the spring problem. A force of 20 pounds stretches a spring 9 inches in an exercise machine. Find the work done in stretching the spring 1 foot from its natural position.

Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the \(y\) -axis. $$ y=x^{2}, \quad y=4 x-x^{2} $$

The integrand of the definite integral is a difference of two functions. Sketch the graph of each function and shade the region whose area is represented by the integral. $$ \int_{-\pi / 4}^{\pi / 4}\left(\sec ^{2} x-\cos x\right) d x $$

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines. \(y=2 x^{2}, \quad y=0, \quad x=2\) (a) the \(y\) -axis (b) the \(x\) -axis (c) the line \(y=8\) (d) the line \(x=2\)

Solve the differential equation. $$ x y+y^{\prime}=100 x $$

In Exercises \(3-14,\) find the arc length of the graph of the function over the indicated interval. $$ y=\frac{3}{2} x^{2 / 3}+4, \quad[1,27] $$

Hooke's Law In Exercises 3-10, use Hooke's Law to determine the variable force in the spring problem. An overhead garage door has two springs, one on each side of the door. A force of 15 pounds is required to stretch each spring I foot. Because of the pulley system, the springs stretch only one-half the distance the door travels. The door moves a total of 8 feet and the springs are at their natural length when the door is open. Find the work done by the pair of springs.

Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the \(y\) -axis. $$ y=4-x^{2}, \quad y=0 $$

In Exercises 9 and 10 , find the area of the region by integrating (a) with respect to \(x\) and (b) with respect to \(y\). $$ \begin{array}{l} x=4-y^{2} \\ x=y-2 \end{array} $$

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines. \(y=x^{2}, \quad y=4 x-x^{2}\) (a) the \(x\) -axis (b) the line \(y=6\)

Write and solve the differential equation that models the verbal statement. The rate of change of \(Q\) with respect to \(t\) is inversely proportional to the square of \(t\).

In Exercises \(3-14,\) find the arc length of the graph of the function over the indicated interval. $$ y=\ln (\sin x), \quad\left[\frac{\pi}{4}, \frac{3 \pi}{4}\right] $$

Hooke's Law In Exercises 3-10, use Hooke's Law to determine the variable force in the spring problem. Eighteen foot-pounds of work is required to stretch a spring 4 inches from its natural length. Find the work required to stretch the spring an additional 3 inches.

Find the area of the region by integrating (a) with respect to \(x\) and (b) with respect to \(y\). $$ \begin{array}{l} y=x^{2} \\ y=6-x \end{array} $$

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the given lines. \(y=6-2 x-x^{2}, \quad y=x+6\) (a) the \(x\) -axis (b) the line \(y=3\)

Write and solve the differential equation that models the verbal statement. The rate of change of \(P\) with respect to \(t\) is proportional to \(10-t\).

In Exercises \(3-14,\) find the arc length of the graph of the function over the indicated interval. $$ y=\ln (\cos x), \quad\left[0, \frac{\pi}{3}\right] $$

Hooke's Law In Exercises 3-10, use Hooke's Law to determine the variable force in the spring problem. Seven and one-half foot-pounds of work is required to compress a spring 2 inches from its natural length. Find the work required to compress the spring an additional one-half inch.

Use the shell method to set up and evaluate the integral that gives the volume of the solid generated by revolving the plane region about the \(y\) -axis. $$ y=2 x, \quad y=4, \quad x=0 $$

In Exercises 11 and 12, determine which value best approximates the area of the region bounded by the graphs of \(f\) and \(g .\) (Make your selection on the basis of a sketch of the region and not by performing any calculations.) \(f(x)=x+1, \quad g(x)=(x-1)^{2}\) (a) -2 (b) 2 (c) 10 (d) 4 (e) 8

Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line \(y=4\). $$ y=x, \quad y=3, \quad x=0 $$

A differential equation, a point, and a slope field are given. (a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the given point. (b) Use integration to find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the result with the sketch in part (a). To print an enlarged copy of the graph, go to the website www.mathgraphs.com. $$ \frac{d y}{d x}=x(6-y) $$

In Exercises \(3-14,\) find the arc length of the graph of the function over the indicated interval. $$ y=\frac{1}{2}\left(e^{x}+e^{-x}\right), \quad[0,2] $$