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Problem 24 Easy Difficulty

Use a graph to find approximate $x$ -coordinates of the points of intersection of the given curves. Then use your calculator to find (approximately) the volume of the solid obtained
by rotating about the $x$ -axis the region bounded by these curves.
$$y=3 \sin \left(x^{2}\right), \quad y=e^{x / 2}+e^{-2 x}$$

Answer

$$
V=\pi \int_{0.772}^{1.524} 9 \sin ^{2}\left(x^{2}\right)-\left(e^{x / 2}+e^{-2 x}\right)^{2} d x \approx 7.518617
$$

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Video Transcript

Okay, so for this problem were asked to find the value. When we have the outer radius as being three sine x squared, our inner radius is going to be e to the X over two plus e to the negative two x. And of course, when you're looking at the graph, the outer race is going to be the one that's above and your earth, the upper boundary of the section that you're finding. Um and then we look at the way the graphs work on this one. Um, is you're kind of looking at this curve and then you're looking at something like this. So we wanna look at the intersection points as our boundaries. So the intersection points are at 1.5 to 4 and at 0.772 So I want to use the washer method because we have two different equations that we were comparing. And so, if you recall the washer method is the volume equals pi with the integral from a to B of our one squared minus are two squared d x. So, looking at this, our volume is gonna be pi from our integral of 1.5 to 4 2.772 of three sign of x squared And then all of that squared minus our Eat X to the are half X power plus E to the negative two X squared DX. And then, of course, utilizing a graphing calculator app, we realized that we have the the volume is being 7.5186