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Problem 3 Hard Difficulty

Find the volume of the solid that results when the region bounded by $x= 1-y^{2}$ and the $y$ -axis is revolved around the $y$ -axis.

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Video Transcript

So for this problem, we know that we're solving for the volume of the read that is created when we rotate. Uh, the equation X is equal to one minus y squared and the y axis about the y axis. And so, in order to make it a little bit easier to figure out the into role that we want to set up, I'm going to go ahead and Graff or equation here eso Since we know that we have, why squared? That means that essentially, this is just going to be a problem and us, and it's negative. Y squared is going to open to the left. Oh, so that means that we are going to have thes three points 1001 and zero negative one autograph and again, our problems just going to open to the left. I'm until our graph is going to look like this, and we also know that it's found by the Y axis. I'm so essentially we are rotating this portion right here about the y axis. So that means that we are going from wise equal to negative one up to you wise, equal to positive one. And so, since our equation is already in terms of wife. We're just going to be able to integrate with D Y rather than DX. And so, uh, like I said, since we're using our wide terms here, this is going to be the integral over our range of why values. So for wise, negative one too Wise, positive one of pi times our radius, which is just our function right here, which is one minus y squared. Do you want? I'm going to square this equation here, since the equation for the area of a circle is pi r squared. Oh, so we can go and start integrating. So first I'll pull out my pie from the equation from the integral and then foil out of this one minus y squared square term. And we're going to get that. This is pi times integral from negative 1 to 1 of ah one minus two. Why squared plus y to the fourth power do you want? And so we can see that using arc our rule This is going to be a pie Times why minus 2/3. Why cubed plus 1/5 lodge of the fifth power evaluated from negative 1 to 1. Oh, and so essentially here we're just going to go and start by playing in our positive one. That's going to be one modest 2/3 times one cube, which is just have one plus 1/5 time's a one on fifth car, which will be times one and then minus. Now we're going to plug in. Y is equal to negative one A. So mine is pi times negative. One minus 2/3 times negative one cube plus 1/5 times negative one cubed, uh, negative one to the fifth Power. Excuse me. And so from here, our last step is just like to be simplifying and combining all of our terms, and we get that the volume of this solid that's going to result is going to be 16 pies over 15.