Problem 35 Medium 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 = \ln (x^6 + 2) $ , $ y = \sqrt{3 - x^3} $

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Wen Z.

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Amrita B.

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uh like right across you're giving curves. And were asked to use a graph to find approximate X coordinates of the point of intersection of these curves. Then we're asked to use our calculator to find approximately the volume of the solid obtained by rotating about the X. Axis. The region bounded by these curves. Well. So the curves are y equals the natural log of X to the 6th plus two. And why equals they swear route of three minus X cubed. That was actually the first draft to find the X coordinate to the point of intersection graph. This both these curves. So here's a graph of both curves and we see that there are two points of intersection. There's one here and one here. X coordinates of these points are approximately yeah, worth Phative 1.467. I love you. And About x equals 1.09. 1 took it and they hated it. My friend said yeah, he was a book now to find the volume of the salt obtained by rotating about the X. Axis. Well if you imagine we're talking about the X. Axis we'll have to use the washer method. Yeah. The whole body. Our outer radius is going to be our top function. Which is Then there were six that was square root of three minus X cubed. No I tried smoking once again. And the inner radius. Mhm. Yeah. You don't remember. Exit is The bottom function. Natural log of X. to the 6th plus two. Yeah. I won't actually that's not true that I was about to say I won't do much. Yeah. Mike you're out on the one hand I actually neglected. There is another X coordinate. If you zoom out further on the graph There's another point about x equals negative. 4.09. Yes. Yeah. One which is also the X coordinate of a point of intersection. It's not showing on this. So therefore we actually have two different kinds of washers. So these are the right washers that have these radio mushrooms can make whatever. Man we should go back to south. Now on the other hand, for the left washer just yeah, now we switch it to the outer radius. Like that is now the natural log of X quit to the sixth plus two. And the inner radius is the square root of three minus X cubed. Considering. Yeah, okay. And therefore are volume V. And totals is going to be the left volume plus the right volume. It's like that's like a thing. This is equal to the integral from X. Is about -4.091 seven To about x equals negative 1.467 of our outer radius. For the left washer Which is natural log of X. to the 6th plus two squared minus the inner radius which is Square root of three -X Cube squared D. X. No so it's pi times this plus pi times the integral from -1467 About to about the point 1.091. And then we have our outer radius is squared of three minus X cubed squared minus our inner radius. Natural log of X to the sixth plus second plus two squared D. X. This is a pretty long and difficult integral. So if he is a calculator you should find that the answer Is approximately 89.023. Right? Wait, how would he? It's really like.