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# Use a computer algebra system to find the exact volume of the solid obtained by rotating the region bounded by the given curves about the specified line.$y = x$ , $y = xe^{1 - \frac{x}{2}}$ ; about $y = 3$

## $V=\pi \int_{0}^{2}(3-x)^{2}-\left(3-x e^{1-x / 2}\right)^{2} d x=24 e-\frac{142}{3}-2 e^{2}$

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Applications of Integration

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### Video Transcript

okay. As this question specified to draw a diagram, we know that we can do Note this to be the axis of rotation and draw our wine y equals acts to indicate this shaded region over here, this tiny sliver. And then we know that our intersection points are pretty straightforward too common to and then the origin zero comma zero. Which means now we know we're using the formula pi times the integral from 0 to 2 intersection points or 00 and 22 out already is squared, which is three months acts minus the inner radius square. And don't forget to square each of these individually. That's whether written in parentheses. Okay, this simple for us to this Or if you want to write this industrial form 9.82476 this is a calculator problems. You can write this industrial form

#### Topics

Applications of Integration

Lectures

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