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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.

$$y=\frac{1}{4} x^{2}, y=5-x^{2} ; \quad \text { about the } x-axis$$

$=\frac{176 \pi}{3}$

Applications of Integration

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find the volume between these equations. Why, it was 1/4 x squared by equals five minus X squared and we're revolving around the X axis. So first of all, it's gonna sketch of these graphs. 1/4 X squared is a wide in shorter version of a problem. So some sort of wide, probable happening here and then five minus X squared is still the probably shape. But it's been flipped about the X axis because of that negative in front. So it's facing down and it shifted up. Five. So 12345 facing down standards, highs proble. So the area we're looking at is this part I'm shading in green. And if I'm going around the X axis, the number evolving here notice there is this space missing between our figure and the axis, which means it's gonna be a washer. So because it's the washer set up, we use volume is pi times the integral capital R squared minus little r squared d X cause I'm going on the X axis and I need my lower X and my larger X excellent x you so as I'm considering what things to fill in the first hand probably should conquer. Here is what are these x values. So x one and x two to find that I'm just going to let the two equations equal. So we take that 1/4 X squared and let equal five minus next win. Well, if I add this expert over, I get 5/4 x squared. And then if I use that reciprocal of forfeits on both sides, X squared equals four. So X is plus or minus two, so I can put that as pleasure minus two year. The other thing that can be helpful with this set up is because I see both parabolas are symmetric about the y axis. I could actually double my and a girl and just go from 0 to 2. I could just do this half of it and then double it. So that is an option here, and especially if you're calculating the integral by hand. It is sometimes easier to use zeros at lower bound and then just double your results. So that is an option. The ours are not gonna change. So if you are working that option, you're gonna still get the same final answer and those ours. You're not gonna change. So let's look for capital are in lower case R And I do think it's easier sometimes to look for the lower case our first, especially on this problem, I feel like it really jumps out. Lower case ours This stuff we're taking away. So the stuff I'm taking away upper minus lower here is the lower put parabola down to the axis. The lower proble is the 1/4 x squared the axis. We're evolving around a zero. So for a little R, I would put 1/4 x squared for capital are its You know what would be the radius if this whole thing were filled in if there was nothing missing. So like if we went all the way from the axis to the top so if nothing was missing, that lower would be zero and the upper proble here is that five minus X squared. So upper minus lower five minus x squared minus C grow. I could destroy this five minus X squared. So now I have the set up for my integral. This is a time where if I can, I would definitely calculate the integral and for the sake of time here at so I'm gonna dio so just evaluating the integral part, I got 176 over three multiplied by pi. I would say I get 1 76 high over three as my final answer. If you had to work this by hand, you would have to expand this binomial here and then use this expo, Nate, and go term by term, it's not too Cabinet definitely takes a lot more time. So if possible, this is a great one to calculate.

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Applications of Integration