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Find the volumes of the solids

$V=\frac{\pi}{6}-\frac{2}{9}$

Calculus 3

Chapter 15

Multiple Integrals

Section 7

Triple Integrals in Cylindrical and Spherical Coordinates

Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

Lectures

04:18

In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. The area above the x-axis adds to the total.

26:18

In mathematics, a double integral is an integral where the integrand is a function of two variables, and the integral is taken over some region in the Euclidean plane.

09:39

Find the volumes of the so…

11:58

10:33

13:28

14:09

The following volume volume is bounded by the following surfaces you have here. We call this is the c axis x and y. The upper part is given by some surface, like the where the surface is given by c is equal to 4 minus 4 times x, squared plus y squared, and have this lower part. The lower part is like some shape like that. That has the coordinates c is equal to x, square plus y squared squared minus 1 point so would like to find what is the volume of that object? What is the volume, so we can use a cylindrical, dricooscylindrical coordinates and the well we'll have. That c is between those 2, and this part here that is r squared x, squared plus y squared, is r squared, so that you do r d, and this c is written, that that is this part, is r squared, so r, squared squared so r to the Fourth, minus 1 point: this is the lower part and then the upper part will be that 4 minus 4 r squared and then we consider our radius r, so that should be obtained for c equals 0. The inequal he this equation we have over this 10 is equal to 4 minus 4 x, squared plus y squared, which gives us 4 equal to that, so that we obtain divided by 4 x. Squared plus y square is equal to 1 and you obtain the center. With s equals 0. We would have 0 is equal to x square plus y squared squared minus 1 point so that this is equal to 1 and then is the same equation so that you obtain it by making it smaller than 1 to obtain this. This scare here so that r is going to go from 0 up to 1 and the there goes all the way around. That'S the whole turn so from 0 to pi. Now there would be 02 pi and we can go ahead and integrate this first. So this integral from ze 2 pi is going to be there evaluated into pi and 0, which is equal to 2 pi. We have a factor: f, 2 pi, there, 2 pi, it's gonna, be 2 pi. This volume is equal to 2 pi times central from 0. Up to 1, the r and then from r square to the fourth minus 1 up to 4 minus 4 minus 4 r squared r d. Well, we can just write it here, so you have that then, if integrate integrating this c, because the rest is a constant with respect to c, so that the interval of this is ucatella. Those those 2 and points would make their integral to be 2 pi times. The integral, from 0 up to 1 of 4 minus 4 r, squared minus r to the fourth power minus 100 times r b r, so that we we can so that we get it. I e minus with minus, becomes plus plus 1 point, so this will be 4 plus 1. There is 5 minus 4 r, squared minus r to the fourth power and that times r, so that this becomes. This would be 5 r, minus 4 r cube minus r. To the 5 and integrating these roll have so the integral 5 hours r is 5 halves of r squared 12 upon 4 r cube would be to simply r to the fourth and in talor to the fifth 6 divided by 6. So these evaluate between 0 and 1, because those are the bounds that we have there and then times 2 pi, so that this would be equal to the volume of that. And then, if we plug in 1, we have what 5 halves minus 1 minus 1 over 6, or you can multiply these to make a condemnator can multiply by 3 here and then. This 1 is 6 over 616 over 6, so that you would have 5 times 3, that is 15 minus 6 minus 1 over 2 times 336 point. So this would be 15 minus 7, which would be 8, so this would be 8 over 2 times 3, but we also have a 2 pi there, 2 pi exact, and so all these 2 councils that 2 there obtained the doe should be equal to 8. Third, i that should be all the volume of this will be able to 8 by thirds.

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