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Problem 18 Medium Difficulty

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.
$$
y=x, y=0, x=2, x=4 ; \quad \text { about } x=1
$$

Answer

$\frac{76}{3} \pi$

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Video Transcript

all right, We're giving a figure. If you look at it in your textbook, you see that it has to find the volume off the solid obtained by rotating the region, bounded by the given curves about the specific specified line. And we want to sketch the region solid and a typical disk or washing. Okay, so basically, all it's really asking us to do is find the volume of the solid when we have executes y because x Y zero executes two to execute four about X equals to what? So when you look at your figure, what you could see is that you can obtain their solid, um, by taking the volume equation to be equal to the integral from zero to, well your area and then adding the region from zero to 24 Well, your area equation okay. Our area we know and be written as the pi multiplied by radius squared, we would take our outer radius first, which in the case of the region from zero to we know is four minus one is our outer region radius. And then we get subtract high multiplied by our inner radius, which would be to minus one and you'll get your area to be equal to a pie. Okay, this is just for this region. Judy too. And then you go ahead and take the integral from 0 to 2 a pie. Why, you will get 16 pie. All right. Now, from the region of 2 to 4, our area equation I'll call this a sub to In the second region which is from 2 to 4 is the same thing or radius Larger radius square times pi minus. We're in a radius which would be in this case, why minus wine. Sweat. Okay. And then when you multiply all of this cross, try to simplify. You will get something that looks like pie eight minus y squared. Plus two y When you expand this whole equation out here right now, when we take the integral from 2 to 4, they were a to y You have shooting four Have pie need to minus y squared plus two y Do you want alright? When you integrate this, you get pie that comes out. You have eight y minus y cubed over three waas two y squared over two Then that is from four to now. When you go ahead and plug before and to in, and you go and solve. For that, you will get 28 over three pie. All right? And then you take your to area to inter girls that you've calculated for, and you add them together to get your volume. Your volume will be the first area that you kept there, which is 16 pie or your first volume calculate. Excuse me. 16 pie in the second volume, calculated for the second region. Just 28/3 high. And if you go ahead and expand this out as well, this comes out to be 76/3. Hi. That will be your final volume there. All right, well, I hope that clarifies the question there. Thank you so much for watching.