Problem 36 Hard 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 = 1 + xe^{-x^3} $ , $ y = \arctan x^2 $

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Wen Z.

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Amrita B.

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Video Transcript

we were given curves. We'll have to find use a graph to find the approximate X coordinates of the points of intersection of these curves. Then rescues are calculated to find approximately the volume of the salt obtained by rotating about the X. Axis. The region bounded by these curves. The curbs are Y equals one plus X times E. To the negative X. Cubed. Yeah. Then all the and why equals arc tangent of X squared. Got a new pitch next to a tiny microwave and warming up but really just suck on your other. So first I'll use Dismas online graphing calculator to graph these curves and to find the points of intersection. Right. Sure. It's a trap. So here is the graph of the curves. The graph appears to have what we see two points of intersection here and here. The point of intersection on the left. This has X coordinate about negative 57 Any second X coordinate on the right. The point of intersection is about 1.391. You are supposed to have more? She's right. But so it's a catered event shot. No. Now if you wrote this about the X axis, we're going to have to use the washer method for our washers. The outer radius. The analyst type is the top function which is one plus X times E. To the negative X cubed. Break out of your. Yeah. And the inner radius was maybe they say oh did you? And groceries channels Z equal to arc tangent of X squared. It's a bit of a like a little treat as everyone knows. And so by the washer method, the volume of the rotated region. The solid is pi times the integral from the left X X 1 to the right x coordinate X two of our outer radius one plus X. E. To the negative X cubed squared minus the inner radius arc tangent of X squared squared dx. This is a complicated volume, but if you use a calculator, you should find this is approximately key does of only your own son, but it 6.9234 less than on sunday. There are more safe. Okay, But so in your ass.