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Use spherical coordinateswhere B is the ball with center the origin and radius 1. Evaluate Il &+7+29- dV,CilLkte WANeed Help?RorontMALaLL...

Question

Use spherical coordinateswhere B is the ball with center the origin and radius 1. Evaluate Il &+7+29- dV,CilLkte WANeed Help?RorontMALaLL

Use spherical coordinates where B is the ball with center the origin and radius 1. Evaluate Il &+7+29- dV, Cil Lkte WA Need Help? Roront MALaLL



Answers

Use spherical coordinates.
Evaluate $\iint_{B}\left(x^{2}+y^{2}+z^{2}\right)^{2} d V,$ where $B$ is the ball with
center the origin and radius $5 .$

We are given a triple integral and rescue spherical coordinates to evaluate this integral. So we're given that the integral is the triple integral over the region B of x squared plus y squared plus z squared squared tv leon. Where B is the ball with center at the origin. There's yeah. And a radius of five. To use spherical coordinates. First rate be in spherical coordinates. So for B we have that it's a ball. So we can simply describe it with a row Line between zero and the radius of the ball. five. As for five and data, there are no restrictions other than file is between zero and pi and fatal lies between zero and two pi. So we can rewrite are integral now as an integral a spherical coordinates. So the integral from 30 equals 0 to 2 pi integral. From 50 to pi integral from 0 to 5 of our function. But in spherical coordinates X squared plus Y squared plus z squared is rho squared squared. And because of the change of coordinates swift multiplied by rho squared sine phi dear! Oh defy decide to she breakfast now we can use for business there and this is a product of integral. We get the integral from 0 to 2 pi D. Data times. The integral from zero to pi of sine phi defy Times the integral from 0 to 5 of road to the 6th D row straight up. Mhm. Taking anti derivatives, we get fada From data equals 0-2 pi times negative. Co sign fi fi equals zero to pi Times wrote of the 7th over seven From Bro Equal 0- five. No This is equal to two pi times negative co sign of Pie which is negative one uh plus one times 5 to the 7th over seven comes from like a country, this is after multiplying everything out. Uh 312,000. The mm 500 times pi Right over seven literally rightness, highest

Okay, So you wanna go ahead and in a great, um, find this double integral? Um, and don't wanna go would be ah x squared plus y squared plus z squared. Squared. Devi. Um, where b is the ball with senator a t origin and read. It's five. So this is center 000 row because five. So don't get confused. There's there's to medical science, but I just shows. Ah, that means you're doing a surface interval. We're gonna have a spherical coordinates. Um, where we have trouble in a goal of expert plus y scruples East Great is roast squared. So this is Rose squared, squared or row to the fourth Power and IV uses roast grid Sign by the roadie by data. And we know that since we have a ball our roses from 0 to 5. R 50 to pi and Arteta 0 to 2 pi. So then we just evaluate this. So we had rose to six power. So you have 0 to 2 pi sort of pie 0 to 5 of road to the sixth sci fi. The road defied data and rose. Six wrote to the six Power gives us the inner integral evaluates to row 7/7 from 0 to 5. Signed by. Do you, uh, the by. So, um, since you're at five, it's five to seventh and 00 We can just bring the five 27 power over seven outfront and continue on with our integral, um has follows. And we know that the interval of sign eyes just native co sci fi. So we'll have this hotter. And a girl still negative co signed by through the pi data and still working on the outside. Um, we'll see that negative co sign Pi is just positive one and then negative. Negative. Coastline zero is positive co sign zero or positive one. So this ends up being too data and, um, two comes outfront, then sustaining 12 constants and into pie. So let me get four pi over seven times 5 to 7, um, as our final answer. And if we're gonna go ahead and write this all out So 57 those four just to of 312,500 Hi. Over seven. So, um, that is our final answer.

To the sphere. Evaluate Double Integral right is mhm. Will be Congo, where B is people motion This five born spiritually corden reaching theme. Wrong. All right, which that well, ball has five. So you will, while between data logic from zero and and by life between zero and time. So the triple integral, Uh, very close once. Mhm, Mhm, right. Illicit. Integral. Who was, um, vote the home. Go from here long. Well, function or the X prospect. Francisco Very scared. Differential becomes Rose for its son. He I mean you. Really? Yeah. Yeah. Means well. Um well, I'm fine. Welcome. Zero to pi. Yeah, Girl from Verify. See you will. Yeah. Mhm. Yeah. Negative. I'm fine. Right? Yeah. All right. And one of the ones, um, bomb evaluating. We get mhm. Mhm. All right. Um, 70,000 25. Simplifying of saying 312,000 500. There's seven times pi. What? Approximately 20,000 49

Okay. We want to integrate X. E. To the X squared plus Y squared plus Z squared over the unit ball in the first quadrant. So let's take care of that first. Okay, pretend like that's the first quadrant. Yeah. Okay. So since roe is X squared plus Y squared plus Z squared. It's starting here at zero and going out To the edge of the ball. So row is going 0-1. Okay? Um E starts here at zero and goes to pi over two because that's the first quadrant equals zero or first octet tuf equals pi over two. And then data will start here at zero and go here to pi over two. Mhm. Okay so that wasn't the hard part. Make the substitution. X's rho sine fi co sign data. E to the rho squared rho squared sine fee. D zero D V. D. Stated. Okay, so now gather up all the stuff you haven't eat, the rho squared. You have a row cubed, You have signed squared fee cause I'm data. The row defeated Saito. Okay, so since the row is not mixed in with all the other stuff, we can just go ahead and pull that integral out and work on it. Okay, so the obvious thing is to let you be rho squared and then you'll need a row to make d'you. So we need to separate this into 0 to 1. Heat the rho squared rose squared row the row. All right. So if you is rho squared and d'you is to roe deer? Oh So we needed to in there? No 1/2. Mhm. And if roe is zero U. is zero square which is still zero If roe is one You is one square which is one. So now we have 1/2 zero is still zero one is still one E. To the U. You deal. So now we have to do integration by parts. So I'm going to let W. Equal you and DV equal eat the U. D. U. So then D. W. Is D. U. And V. Is eat the you. So now we get 1/2 W. V minus the integral VW. So that's one half U. E. To the U minus E. To the U. From 0 to 1. That's one half one each of the one minus eat the one minus zero minus eat to the zero. So those canceled because they strapped and you get one half times minus minus one so one half. Okay so now we can put that in 0 to 10 to 11 half. Sine squared fee. Pro sank to to D. V. D. Theater. Okay now we're gonna have to put in an identity for this. I'm gonna put the one half out in the front here, 1- Cosine to fi over to cosign theta. Do you feed the data? So I'll put this to out in the front so now I have 1 4th here to one. Okay the integral of one with respect to fee is fee minus the integral of co sign to fi we need a two in there and a one half. Okay so that we can make the fee or do you if we let you be to fee. So we get one half interview of co sign. Is the sign to feed. Um Oh these are not 0-1. This is pi over two sorry 0 to Pi over two. So now we have 1 4th 0 to Pi or two. However 2 -1 half the sine of pi which is 0 -0 0. So now we have pie over 80 pi over two integral Cosign Theta. D. Theta. The integral of the coastline is the sign From 0 to Pi over two. That's pi over eight. Well minus zero By over eight. I get.


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