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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 solid enclosed by the cardioid of revolution $\rho=1-\cos \phi$

$8 \pi / 3$

Calculus 3

Chapter 15

Multiple Integrals

Section 7

Triple Integrals in Cylindrical and Spherical Coordinates

Missouri State University

Harvey Mudd College

Baylor University

University of Nottingham

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.

05:18

(a) find the spherical coo…

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14:10

08:22

In Exercises $55-60,$ (a) …

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Okay. So India's problem you're asked to find first to find the limits. So part A. Is asking us to find the limits For the integral that calculates the volume of the given solid. The solid here I'm calling the solid. He is the target of revolution given by the equation girl equals 1- constant be. So I attached a sketch of how this looks like in this case in the case that you are not familiar with this. So let's take a look at the fee values. So it should be clear based on you know the diagram or discussed that you have on the right side that the fee values will start at fecal zero. Sophie equals zero is on the positive zP access all the way two the negative Z axis. So they're the fee value here is has a value of pi so we know that he would range from zero all the rich pie. What about data? Some data by looking at our sketch data will do something like this. So you start somewhere here and you go around. So you first go towards the positive why access is starting at positive X axis. You go around this only following until you come back to where you started. So when you started with is data equals to zero and you come back to the same point. But the data value the area is going to be equal to pi. So the range violence for data will go from zero all the way to die. Finally the range of others for rope. Well we're going to start at the origin of the earth. The value of row is going to equal to zero and we're going to be you know all the way to the actual cardio. It which with this equation role equals one minus possibility. Several will start at zero and it will end at one minus course in the feet. So these pretty much are the limits for fee Data and Row. So that's part one and for part B. And he asked us to evaluate the integral for the volume. Pretty much. So. First the volume of this object which we're calling E is defined to be the triple integral over the boundaries of this only E Of one TV. We're using spherical coordinates. So we already know that Data will go from 0 to Pi. We already know that fee will go from syrup. I And then we will go from 0 to 1- course Sanofi. And um here we are going to want times DV in spherical coordinates which you know in Britannia recording DZ Dy dx in any order in spherical coordinates. This will become a real squared times sine of the the role, the fee vita. So with that in mind we can now start by a volley industry only integral. So we can start by noticing that we can evaluate the anti or constricted pie of one of the data times the double integral first integral high And the integral from 0 to 1- closing feet of rho squared times Sanofi the road defeat. So the first integral on the left side should be straightforward, has a valid high. Then we're going to multiple ideas by the anti cruise ship. I and the integral with respect to grow of rose grey Israel cuba were three. So we're going to have Grow to the 3rd power Over three times sine of feet. And you're bothering this from real Echo 0 to roll because one man is causing a fee and then whatever we get, we still have to continue with the next integral. So this is going to give us two pi. I can pull out the one third. So I'm gonna have to pirate three Integral from 0 to Pi. Of uh one minus course sign of B. Well obviously the third power times sign of the defeat. And how do we value it? Is integral. We can use a user decision. So let You be equal to one minus school sign of feed. The you is going to equal to positive sign of the defeat. The range follows for you. Um you will range from 0-2. So this integral will now be two pi over three times the integral from zero to with respect to you remember that. And one minus causing a fee is you. So these youtube third power. The you. So what is this equal to? Well, now this is just going to equal to two Pi over three. And um to the fourth, Power over four, evaluate this expression from 0 to 2. And all of these will just give us two pi over three times four. So all of these food just give us a pie five x 3. So that gives us the solution for this problem.

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