Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer.

$ y = x^3 $ , $ y = 0 $ , $ x = 1 $ ; about $ x = 2 $

$V=\frac{3 }{5} \pi$

Applications of Integration

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Numerade Educator

Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

We're finding the volume of solid obtained by rotating the region bounded by y equals X cubed y zero X equals one about X equals two. Here's our axes like als excuse. It's pretty simple. It looks like that and draw out the other side, too. The y equals zero was even more simple. X axis and X equals one is a very good line. The X equals two, which is the line we're rotating about. He's another vertical line. This is the region that we're rotating. We're rotating about this line, so we're creating Washington. Um, it is important to note that this point where here is known, it's one one because one cube is still one. Now we need a first notice that, uh, the curved grading The larger radius is, though why equals execute line and lie in creating a small radius is X equals one. So first, when you find the distance between the actually close to line in the y equals execute mine. And so we are subtracting X values. So we want to first express this over here in terms of X. That's just why to the one third or the cube root of Why now? The distance itself Cuba of why gives you this distance right here from X equals airline to any point on the curve. What we want is this distance. What we have is this distance and this distance. So to find the distance between the curve in the line, the bigger radius is just to minus X value, which is to minus y cute are y to the one third smaller R is pretty simple. We just need this distance or here. And that is constant regardless of why. Very simply, articles to minus one equals two No. One. And now we have that we can come up with the cross sectional area function yes, equal to the larger areas or a large area. Mice smaller area. Um, plug it in. You gonna pull out the pie? Not by this Out right here. Two squared is for two times two is also four. So you've been for while I'm there, lying to the one third squared is what, two thirds? Oh, now we're still subtracted. One because one squared is just one that becomes three. So that is our cross sectional area. Didn't go ahead with the unicorn. Now now, if you nose here were integrating from Michael zero two white with one because that's how the region spans. Keeping the pie out and we're plugging this in double check, if that's correct. Yep, So we can now Val away inside reviews and the truth of the story History. Why anti derivative of why to the one third is well, one third plus one is why to the four thirds we know that isn't there. Now we normally divide by four theories, but that's just equal to multiplying by reciprocal just three fourths. But we're still multiplying by the four, so these two cancel. So what we end up with is just a three in the beginning. Now two thirds plus one is five carats. We're divided with five thirds, but that's just multiplying by their simple, which is three fits. We don't really care. One y equals zero because one y equals zero. This entire thing is just zero. So we only care when y equals one. When we plugged that in, you get pi times for me minus three plus three fists. And so as our finance, Sir. This three pie or five that is our finalists