Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

(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 between the sphere $\rho=\cos \phi$ and the hemisphere $\rho=2, z \geq 0$

$\frac{31 \pi}{6}$

Calculus 3

Chapter 15

Multiple Integrals

Section 7

Triple Integrals in Cylindrical and Spherical Coordinates

Missouri State University

Campbell University

University of Nottingham

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.

14:10

(a) find the spherical coo…

13:31

12:26

02:18

In Exercises $33-38,$ (a) …

09:54

So we have this region. Is it's a volume in 4 dimensional space? That is the vision in out of the ball given by new spherical coordinates. This ball is described by rho equals to cos f, and we want to compute, so they volume out of that. Inside of the hemisphere that is like this half wall sphere, that is so this is her described by rho- is equal to 2 and c is bigger than 0. So you would like to complete the volume inside these 2 regions, some in between and so for that we would like to do an integral into an imospherical spherical coordinates. So all these region are radicals cos. It can be seen that you have here row this angle and f this angle and broad the distance so in the plane in this plane. That is sent rows the distance and is the angle we're going to have this sphere in the row f plane where fees measured from bar from the c axis. So we have the circle and then we rotate around vera around vera around 2 pi. To obtain these. This circle, this sphere, this sphere, that has diameter 1, and so for this volume you will do as you can see. Our f is going to obtain this circle. You let wolf go from 0, so from 0 up to the, which is a tangle that is piaso. He goes from 0 up to by half, and then we would let thera go all the way around. So there goes from 0 up to 2 pi to give us the whole turn and obtain that region, and all the bound for row is that it has to be bigger than this cost and it has to be smaller than 2 and about you cannot easily be. Let f go from 0 up to perhaps you won't get we'll get these points up to there and we will not get any point below the x right plane. So this is the correct range for so rather this integral would be all the volume will be equal to all. We have the volume element squared sine of the then the ogs from cos p up to 2 f goes from 0 up to pi halves, 0 up to by halves, and then theta goes from 0 up to 2 pi. So not is here done over dentro. Nothing depends on so we can do that integral separately integral from 0 to the would be just thetable, 2, pi and 0. So this will be equal to just 2 pi. So we can, you can forget about antonia factor of the pie, so it will be equal to 2 pi times all the integrals and then for the integrating row first, so the integral. How squared is equal to rho cube? Third, so it would be that evaluated at 2 and cos of thee so that we obtain also a third from there metre from 0 to by halves of low square of 2 cube 2 cube minus this cube minus cos, t cube and bat multiplied by sine of Sine of t b d, we we just left with that interval and here we can do a substitution. If we let u to be equal to cos, then over the side we have u cube, and so we, if, u is equal to cos p, would be minus sine f f, which is almost there so that this integral would turn into the integral 2 pi. Third. 2, cube: minus? U cube du or minus d? U because a sine p d p is equal to minus d? U, which is what we have there so and then here we would have the bonds for you at 0 and you at by halves. So you at 0 will be cos of 0 cos of 0, which is equal to 1, and yet by half this cos of by half this number is 0. So what we can do here is we can flip the intervals and wouwou get a minus so that we we make that minus go away and we flip intervals, so the interval from 0 up to 1 of 2 cube minus. U cube du! Well! The interval of this, so the integral of that will be so we 2 pi third and then times 2 to the star power integral is just and the integral. U cube will be minus theta of minus? U cube minus! U to the fourth power divided by 4 and that evaluated between 0 and 1, so that these would be equal to e plugging in 1, there so 2 pivert of 2 cube times 1 minus 1 to the fourth power divided by 4, and that minus 2 cube Times 0 minus 0 to the 4. He left port. So great, this part is 0 and then- or we have this 2 c equal to 8 point. So you go to 8 minus 1, for this number is equal to 8 times, 4 minus 1 over 4 and at times for it that is 16 times 2, it's 32 to 32 minus 1, that is 31 point or 2 pi. Third, so this number will be 32 minus 131 to 2 pi times 31, fourth divided by 3, and these councils with that, so that we have 31 pi over 2 times 3 over 6. So this this should be. This should be the volume to pi over 6.

View More Answers From This Book

Find Another Textbook

01:58

In each of the following exercises, solve the given inequality.$$(3 x+5)…

03:31

In each of the following exercises, solve the given inequality.$$x(2 x-5…

02:17

Find the unknown.$$\frac{1}{8} x^{2}+3=0$$

05:07

Solve for the unknown, and then check your solution.$$0.2 x-0.3(5-2 x)=0…

00:46

Find the unknown.$$w^{2}=-36$$

02:57

Solve for the unknown, and then check your solution.$$\frac{3}{5} y=9$$<…

02:26

Find the unknown.$$\frac{3}{4} y^{2}+24=0$$

01:45

Find the unknown.$$(2 y-3)^{2}=49$$

01:16

Complete the square in each of the following by putting in the form $(x+B)^{…

01:06

Find the unknown.$$x^{2}+25=0$$