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Find the volume of the region bounded below by the plane $z=0,$ laterally by the cylinder $x^{2}+y^{2}=1,$ and above by the paraboloid $z=x^{2}+y^{2}$.

$\frac{\pi}{2}$

Calculus 3

Chapter 15

Multiple Integrals

Section 7

Triple Integrals in Cylindrical and Spherical Coordinates

Missouri State University

Baylor University

University of Michigan - Ann Arbor

Boston College

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.

08:03

Find the volume of the reg…

06:51

12:22

12:53

Find the volumes of the re…

So we have the following fine region- you have here all these. There have here that x or y, the c axis, and we have a region that is with this. Has the upper bound to be this parabaloid that this probably here is described by c, is equal to x, square plus y squared and then the lower bound is this plane c is equal to 0 very nice? We have that and then, let's all that is inside of the ceiling there of radius 1, so that we have a c doing that here of radius 1 that cuts so that we're going to obtain a region like it is shown there. So this region is called so you want to find what is the volume? So if you do, cylindrical coordinates where we plot here s an hour, the distance to the c axis. We notice that this it has to be like this, where this is the line. Articles 1 is equal to this line and we have this region this or this area, and this this is, that is r squared. So this would be the line. C is equal to r square so that we have that picture picture, and then we wetted that around there to obtain that. So what were we could do? This is that we could say what c is going to be between 0 and r squared and then ho r goes from 0 up to 1 or goes from 0 up to 1 and theta goes all the way around. So all the way from 0 to 2. He would go on between 0 and 2 pi so that this volume would be equal to silintegral coordinates we're going to have r for the voluneerdr, so c goes between 0 and r. Squared r goes from 0 up to 1 and there goes from 0 up to 2 pi, so first we're integrating this, because the rest is constant. The ejus evaluated between 0 and r squared this should be r, squared or so within. It is 0 and r. Squared is just gonna give us our square will be r squared minus 0, so you'll have 1 r times r, squared d r from 0 up to 1. You have this integral, which would be well the interloper cube, so this integral is equal to r to the fourth power divided by 4, so that this between 1 and 0 will be just equal to 1, or so that this is going to be 1. Fourth drop from 0 to 2 pi, the tea, but this integral is just theta evaluated between 2 pi and 0 times 1, which is equal to 2 pi minus 0 divided by 4, so that this cancels and gives us for that is to their supply halves. That is going to be the area the volume the volume is 5 halves.

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