Let A greater than B. If the ellipse shown in the picture is rotated around the x-axis, one obtains the ellipsoid E sub x with volume vol of E x. If the ellipse is rotated around the y-axis, one obtains E sub y with volume vol of E y. Which of the following statements is true? Okay, so let's just calculate the volume of this ellipse because if we can find the volume then we can probably answer all of these questions. So the equation of the ellipse is x squared over b squared plus y squared over a squared is one. That's just from the image given. From this we can solve for y and we can solve for x. And then we can calculate the volume of E x and E y using like two horrible integrals. And honestly I don't want to do either of those integrals. So let's not. And let's figure out a better way to calculate the volume with without doing all this horrible math. So the volume of a sphere with radius r is four thirds pi r cubed. And this ellipsoid that we get is kind of just looks like a stretched out sphere. Like if we take a sphere and then stretch out one of the axes by some amount we should get this ellipsoid right here. Okay, so imagine plugging in r equals one and just getting four thirds pi. Now if we stretch out one of the axes by b and the other axis by a, we just need to figure out how much to stretch the third axis by to get either E x or E y. Okay, so if we rotate around the x axis that means we take this vertical line and kind of spin it in the y z plane. That means we have two axes of our sphere of size a. Okay, so that means that we take our v of r, plug in one, and then multiply it by a squared b because we take two axes and stretch them by a and the last axis is stretched by b right here. Similarly, rotating the other way just multiplies by a b squared because this axis gets stretched by a, this axis gets stretched by b, and we spin this axis around essentially copying it over onto the z axis. So that also gets stretched by b. And now v of one, actually it doesn't even matter what v of one is. We can now use this a greater than b to figure out which one of these is bigger because they're certainly not the same. So if I divide one by the other I can see that in here I get, if I divide the first one by the second one I get a over b, and a over b is positive because a is greater than b. So that means volume of E x is greater than volume of E y, and so our answer is c.

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