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Problem 23 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.
$$y=2+x^{2} \cos x, \quad y=x^{4}+x+1$$

Answer

$$
\approx 23.7802
$$

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Video Transcript

Okay, So for this problem, when you're asked to use the appropriate pieces to be able, Thio, find the volume. Given the fact that we have an outer radius of two plus X squared cosine X, we have an inner radius, which is going to be X to the fourth plus X plus one. And then if we look at the graph, we have X intercepts of approximately negative 1.28 78 I'm sorry, 79 and 0.8843 And so what we want to do is we want to set up our volume equation, so we're gonna be using the washer method for this. So if you recall, the washer is the volume equals pi from A to B of, um, are one squared minus are two squared d x. So we already have our our one in our are two. So are are one is simply just outer radius R two is inner radius and we're gonna be using the's um X, the X intercept. So 1.2879 So we're gonna have to plus X squared co sign X squared minus X to the fourth plus X plus one squared DX. And of course, we're using a at feature on the graphing calculator, so we find out that this is approximately 23.782

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