650
Chapter 6 | Applications of Integration
Rule: The Washer Method for Solids of Revolution around the y -axis Suppose u ( y ) and v ( y ) are continuous, nonnegative functions such that v ( y ) ≤ u ( y ) for y ∈ ⎡ ⎣ c , d ⎤ ⎦ . Let Q denote the region bounded on the right by the graph of u ( y ), on the left by the graph of v ( y ), below by the line y = c , and above by the line y = d . Then, the volume of the solid of revolution formed by revolving Q around the y -axis is given by V = ∫ c d π ⎡ ⎣ ⎛ ⎝ u ( y ) ⎞ ⎠ 2 − ⎛ ⎝ v ( y ) ⎞ ⎠ 2 ⎤ ⎦ dy . Rather than looking at an example of the washer method with the y -axis as the axis of revolution, we now consider an example in which the axis of revolution is a line other than one of the two coordinate axes. The same general method applies, but you may have to visualize just how to describe the cross-sectional area of the volume. Example 6.11 The Washer Method with a Different Axis of Revolution Find the volume of a solid of revolution formed by revolving the region bounded above by f ( x ) =4− x and below by the x -axis over the interval [0, 4] around the line y =−2.
Solution The graph of the region and the solid of revolution are shown in the following figure.
Figure 6.24 (a) The region between the graph of the function f ( x ) =4− x and the x -axis over the interval [0, 4]. (b) Revolving the region about the line y =−2 generates a solid of revolution with a cylindrical hole through its middle.
We can’t apply the volume formula to this problem directly because the axis of revolution is not one of the
This OpenStax book is available for free at http://cnx.org/content/col11964/1.12
Made with FlippingBook - professional solution for displaying marketing and sales documents online