Use the general slicing method to find the volume of the...

Use the general slicing method to find the volume of the following solid. The solid whose base is the triangle with vertices (0,0), (11,0), and (0,11) and whose cross sections perpendicular to the base and parallel to the y-axis are semicircles Set up the integral that gives the volume of the solid. Use increasing limits of integration. Select the correct choice below and fill in the answer boxes to complete your choice.

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00:02 All right, we have a triangle in a 0 -0 -11 and 011 or its vertices.

00:12 So it's a nice right triangle here.

00:15 And then if we cut it up into pieces parallel to the y -axis, then the shape of each one of these is a semicircle.

00:30 Okay.

00:33 And so then the volume is going to be the sum of the areas of those semicircle.

00:40 So let's see, area of a semi -circle is half the area of a whole circle, half pi r squared.

00:53 So we've got to find out what to use for r.

00:57 So what are we going to use for the radius of each of these circles? okay, well, you can see that that's some y value, so it's one -half pi y squared.

01:14 Okay, but the thickness of each circle is dx.

01:21 So we're going to have to figure out what y is in terms of dx.

01:25 Okay, but here's what we're going to get.

01:27 The volume is going to be the sum of the areas.

01:36 Except we're not really going to sum.

01:38 We're going to do fancy summing.

01:42 Okay, so we're going to integrate from 0 to 11, 1⁄2 pi y squared.

01:50 Because that's the area of each one of them, but their thickness is dx.

01:59 Okay, so what y am i talking about? well, oh, oops.

02:04 Anyway, i'm not, oh, i'm a mistake.

02:11 Okay, here is y from here to here, from here to here, from here to here...

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