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Problem 34 Hard Difficulty

Set up an integral for the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Then use your calculator to evaluate the integral correct to five decimal places.

$ y =x^2 $ , $ x^2 + y^2 = 1 $ , $ y \ge 0 $

(a) About the x-axis
(b) About the y-axis

Answer

a) $V=\pi \int_{-0.786}^{0.786} 1-x^{2}-x^{4} d x \approx 3.544596$
b) $V \approx 1$

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Video Transcript

it's were given a region founded by curves and a line and were asked to set up an integral for the volume of the solid obtained by rotating this region bounded by these curves about this line Then rescues are calculated to evaluate the integral correct to five decimal places. The curves are y equals x squared and x squared plus y squared equals one. Oh yes where why is greater than or equal to zero? This is going to be The Parabola and the upper half of a circle of Radius one. And in part a the line is about the X axis like that. Mhm. Black Cat December 22 Central we were playing HFS's Yeah we're putting up opening for jimmy's chicken sack, Coachella. And so we're essentially rotating a thin vertical strip. Do A D. C. Did part about the X axis. We get a washer in other words with inner radius various X squared and an outer radius. Yeah. Well if you solve the equation for the circle and take why to be positive, this is the positive square root of one minus X squared. Oh, your will and grace. So we use the washer method and the volume V is pi times the integral from uh the okay, X coordinate of the intersection of Y equals x squared and x squared plus Y squared equals one. We'll we'll do this first actually. So when are these intersecting? Well, this is when Y plus. Okay. Just taking chest john great equals one. So solving for why we get Mhm. What? Well, we really don't want to find why we want to find X. So instead, yeah, we have X squared plus X squared squared equals one. So we have X to the fourth plus X squared minus one equals zero. And solving we get that X squared is equal to negative one plus or minus the square root of one plus four. All over to This is negative one Plus or -15/2. However, X squared is positive. Seriously considering one plus Route five you too and therefore X is going to be plus or minus the square root of route five minus 1/2. And if you have to calculate this is approximately point 786 Yeah, that fits Okay, so our volume V is pi times the integral from our left point of intersection which is negative square root of route five minus 1/2 2. Positive square root of route five minus 1/2. The of our outer radius squared. Mhm. This is the square root of one minus X squared minus our inner radius X squared squared d x £5000. And so simplifying this is pi times the integral from negative square root of route five minus 1/2 Square root of route 5 -1/2 of this is one minus X squared. Mhm. Yeah. They all do that voice. Yeah. Sorry this woman is x squared this is one minus X squared minus X. To the fourth dx. This is the exact volume. However if you plug this into a calculator This is approximately .354 four. Mhm. Yeah. Always running. Sorry. 3.54459. Yes. Yes. Dude, I love it. Yeah. When they even approach a joke Yeah. Six. Which if we're rounding 25 decimal places. This is 3.54460. Mhm. Of Oh no. Mhm. They're like I like like all right so that's part a in part B. Were asked to use the same curves but now we're going to reflect a rotation, say about the y axis. Yes that way. In the Greater Fort Wayne. Yeah. Now we're rotating a thin vertical strip about the Y axis. So we get a cylindrical shell. It says Veterans first response with the radius of X. Those flags and the height publishers which is you solve the circle square root of one minus X squared minus X squared Yeah. Try to carry. Therefore our volume is equal to using the shell method. The integral from 02 x equals our X intercept are right. Most X intercept This was the square root of route 5 -1/2 mm. This is the same as from part A of two pi times our radius which is X times are height which is the square root of one minus X squared minus X squared dx kids. Yeah. For me. Mhm. Yeah, for America. Mhm. So this is the exact value of our volume, right? However you can use a calculator to find an approximate volume. If you do. The volume is approximately among trying to extract kind of branches weighing down his shoulders. He's just yeah. Yeah. Most Spy comes .9999. Mhm. Eight. Yeah. Two. Yeah. That would be from yeah. Yeah. Sure. Is there a guy who that's his? Yeah. Have a sturdy base your friend couldn't have. He would be your deal. Last list.