Problem 6 Easy 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.

$ 2x = y^2 $ , $ x = 0 $ , $ y = 4 $ ; about the y-axis

Answer

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Video Transcript

we're finding the volume of a solid obtained by rotating the region bounded by two X equals y squared X equals zero and white because four when we're rooting this about the y axis, So let's first began by drawing the region. So here is the X and Y axis. The X equals zero line is very simple. It's just the wind dances and the wine Ingles four Line is also pretty simple. It's just when y equals force and weaken denote that as the line where y equals four and drop back across. Now the hardest part is to X equals y squared, and there are two main ways to think about this. In my opinion, the easiest way to think about this is the first expressed X in terms of why so in this case, you can divide by two on both sides, and you get X equals y squared over two. No, and if you remember, why equals X Square typically looks like this looks like this kind of curve. So if you make it other way, if it's a y equals X squared or X equals y squared, then it should look like this. There's no matter what? Why is actual always be positive? And so the divide by two doesn't really do much. It basically just makes it compresses the curve in half. So when we drove out here because this is just a sketch, we can just draw like this. And now we know what the overall region looks like. So the region bounded by thes three curves slash lines. This is marked in red and we're rotating about the y axis because we're rotating about the y axis, We want to find the cross sectional area function. So when we wrote about the y axis, you'll notice that this is the radius were forming circles that we stack on top of each other. So this cross sectional area function will be a function of why? Because as why changes in this case as why increases from zero before the radius will also increase. Therefore, the cross sectional area will also increase, so we can express in terms of why and because we're making circles its pyre squared are in this case is just ex so as equal Tio y squared over two, as we calculated previously on when we plugged that in, we get a Why is equal to pi times y to the fourth over four. He's were squaring our values. So now we can go ahead and do our animal. We're integrating from y equals zero to y equals four and we're putting in the cross sectional area function that we found before. I'm going to pull out the pie in the over four part because both of those values don't have any relationship to the Y itself. And so what we're left with is just y to the fourth do you want and you can just go ahead and back with this. The anti derivative of wine to the fourth is why to the fifth divided by five. So we can just eat pi over four times. Why? To the office two five zero four from when we plugged the Senate, we get zero zero to the anything. It was just your so we can just get rid of that value. Well, we don't really care about it. We only care about the y equals four Apartment four. And when we all calculate this out, we get two five, six pie over five, and that is our final answer.