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(a) find the spherical coordinate limits for the integral that calculates the volume of the given solid and then(b) evaluate the integral.The upper portion cut from the solid in Exercise 35 by the $x y$ -plane

Calculus 3

Chapter 15

Multiple Integrals

Section 7

Triple Integrals in Cylindrical and Spherical Coordinates

Campbell University

Oregon State University

Baylor University

University of Michigan - Ann Arbor

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.

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(a) find the spherical coo…

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In Exercises $55-60,$ (a) …

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in this problem we are asked to find the hysterical coordinate limits for the integral to calculate the volume of the given solid so far apart. A I've given you a sketch of what the solid looks like on the right side on, based on that. Um Take a look at the data. Data is going to start at the positive X. Axis and you continue to increase the value of data until you come back to where you started at. So you started at data having the value of zero. And by the time it comes back to At that point data will have a value of two pi. So the limit values for data theater would range from is in order to pie for fee. When we started the positive we started on the positive Z axis. The F. B. Has a value of zero and you continue to increase this until you get to the xy plane on the X. And Y plane. He has a value of five or two. So the range values for the variable fee will be from zero all the way to power to for ro ro the limits the role will start up zero until you get to one my nose. Of course I know food so the volume of a solid E. Therefore be the conservative supply. And the truth is, you're the pilot too. And the girlfriends, you're the one mother's causing a fee of one row squared sci fi hero defeated data using skirt records for part B. Now we proceed to evaluate um this triple integral. So we're going to have the volume of this object is going to be anything more conservative supply of one. The data times the integral from zero to pi over two integral from 0 to 1 minus close to my feet. Both rho squared sci fi. The row defeat. Yeah. So the first thing to go on the left side that's just gonna give us to pine And we're going to have the meeting considered a part two. This integral is rolled to the third power the bottom three times Sanofi evaluating this expression from For a week. zero heroic response is causing a fee and then the fee at the end. So these will just come to pi times the integral from 0.2 of 1 -3 sign. We'll see To the third power divided by three times sign it defeat. Um So we can pull out one third from this. So we get to pie by three. So in three rooms near the power to one minus cause you know feed all of this to the third power times sine of the tv. We can make a U. Substitution that U equals one minus causing coffee. Do you will become a sign of fi defeat. The new range values for you will now be from 0-1. And now we can continue this By writing to part of the three times Now. The integral with respect to you. Which arrangement 01 Of you to the 3rd power do you? All of these will be two out of three Times U. to the 4th power divided by four evaluating that expression from coast to coast one. So all of these will be two pi divided with three times 1/4. So we simplified this out here. Just go to the end of the 56 you answered for the volume off the object.

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