91Ó°ÊÓ

Rectifiable Curve Describe the condition for a curve to be rectifiable between two points.

What is the geometric interpretation of the area of the region between two curves?

Comparing Methods What is the relationship between the disk method and the washer method?

Arc Length Explain how to find the arc length of a function that is a smooth curve on the interval \([a, b] .\)

Finding the Volume of a Solid In your own words, describe when it is necessary to use more than one integral to find the volume of a solid of revolution.

Area Between Intersecting Curves Explain why it is important to determine all points of intersection of two curves when finding the area of the region between the curves.

A force of 250 newtons stretches a spring 30 centimeters. How much work is done in stretching the spring from 20 centimeters to 50 centimeters?

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 1 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.

Six joules of work is required to stretch a spring 0.5 meter from its natural length. Find the work required to stretch the spring an additional 0.25 meter.

Finding the Volume of a Solid In Exercises \(13 - 16\) , find the volumes of the solids generated by revolving the region bounded by the graphs of the equations about the given lines. $$y = 2 x ^ { 2 } , \quad y = 0 , \quad x = 2$$ $$\begin{array} { l l } { \text { (a) the } y \text { -axis } } & { \text { (b) the } x \text { -axis } } \\ { \text { (c) the line } y = 8 } & { \text { (d) the line } x = 2 } \end{array}$$