Volumes

Rotating a function on an axis creates a solid. The volume of this soid can be calculated through calculus. Buy using the following equations you can calculate the volumes of solids rotated around any axis, x or y.

The second step in calculating the volume of a roated funtion is determining if it is a solid disk or a washer. If there are no parts on the rotated function that are hollowed out then it is a solid disk. When calculating volumes of solid disks you use the following equation: An area of a function is a washer when there is a section that is hollowed out. When caluclating the area of a washer you use the eqation:

**//Example 1 // **Determine the volume of the solid obtained by rotating the region bounded by  ,   ,   , and the //x //-axis about the //x //-axis. **//Solution // ** The first thing to do is get a sketch of the bounding region and the solid obtained by rotating the region about the //x //-axis. Here are both of these sketches. <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">Okay, to get a cross section we cut the solid at any //<span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">x //. Below are a couple of sketches showing a typical cross section. The sketch on the right shows a cut away of the object with a typical cross section without the caps. The sketch on the left shows just the curve we’re rotating as well as it’s mirror image along the bottom of the solid. <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">In this case the radius is simply the distance from the //<span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">x //-axis to the curve and this is nothing more than the function value at that particular //<span style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">x // as shown above. The cross-sectional area is then, <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"><span class="MPPopup" style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; height: 40px; text-decoration: none; vertical-align: auto; width: 528px;"> <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">Next we need to determine the limits of integration. Working from left to right the first cross section will occur at <span class="MPPHSpan" style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 0px; height: 1px; position: relative; text-decoration: none; vertical-align: auto;"> and the last cross section will occur at <span class="MPPHSpan" style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 0px; height: 1px; position: relative; text-decoration: none; vertical-align: auto;">. These are the limits of integration. <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;">The volume of this solid is then, <span style="background-color: transparent; color: #000000; display: block; font-family: Times New Roman; font-size: 16px; text-align: left; text-decoration: none; vertical-align: auto;"><span class="MPPopup" style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 16px; height: 247px; text-decoration: none; vertical-align: auto; width: 385px;"> <span class="MPPHSpan" style="background-color: transparent; color: #000000; font-family: Times New Roman; font-size: 0px; height: 1px; position: relative; text-decoration: none; vertical-align: auto;"> media type="file" key="Volume Known Cross Sec Y Axis 2009 A.mov"media type="file" key="Test

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