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Volume By Cylindrical Shell about the Axis Suppose is a continuous function in the interval and the region is bounded above by and below by the axis, and on the sides by the lines and If is rotated around the axis, then the cylinders are vertical, with and The volume of the solid is given by Volume By Cylindrical Shell about the Axis Equivalently, if the volume is generated by revolving the same region about the axis, then the cylinders are horizontal with where and The values of and are determined in context, as you will see in Example 6.

The shape of the graph is called the astroid because it looks like a star. The equation of its graph is The figure below shows a suspension bridge. The cable has the shape of a parabola with equation The suspension bridge has a total length of and the height of the cable is at each end. Show that the total length of the cable is Review Answers . Area of a Surface of Revolution Learning Objectives A student will be able to: Learn how to find the area of a surface that is generated by revolving a curve about an axis or a line.

26 The Arc Length Problem If is a smooth curve on the interval then the arc length of this curve is defined as Example 1: Find the arc length of the curve on (Figure 18). 27 Solution: Since Using the formula above, we get Using substitution by letting , then Substituting, and remembering to change the limits of integration, Multimedia Links The formula you just used to find the length of a curve was derived by using line segments to approximate the curve. The derivation of that formula can be found at Wikipedia Entry on Arc Length.

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