91Ó°ÊÓ

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

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

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. $$ y=x^{2}, \quad y=4 x-x^{2} $$

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 \(x=2\)

Pumping Water A rectangular tank with a base 4 feet by 5 feet and a height of 4 feet is full of water (see figure). The water weighs 62.4 pounds per cubic foot. How much work is done in pumping water out over the top edge in order to empty (a) half of the tank? (b) all of the tank?

Sketch the region bounded by the graphs of the algebraic functions and find the area of the region. $$ y=\frac{1}{x^{2}}, \quad y=0, \quad x=1, \quad x=5 $$

In Exercises \(27-30\), use the disk or the shell method to find the volume of the solid generated by revolving the region bounded by the graphs of the equations about each given line. \(y=x^{3}, \quad y=0, \quad x=2\) (a) the \(x\) -axis (b) the \(y\) -axis (c) the line \(x=4\)

(a) use a graphing utility to graph the region bounded by the graphs of the equations, \((b)\) find the area of the region, and (c) use the integration capabilities of the graphing utility to verify your results. $$ y=x^{4}-2 x^{2}, \quad y=2 x^{2} $$

Determine which value best approximates the volume of the solid generated by revolving the region bounded by the graphs of the equations about the \(x\) -axis. (Make your selection on the basis of a sketch of the solid and not by performing any calculations.) \(y=e^{-x^{2} / 2}, \quad y=0, \quad x=0, \quad x=2\) (a) 3 (b) -5 (c) 10 (d) 7 (e) 20