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Let $D$ be the smaller cap cut from a solid ball of radius 2 units by a plane 1 unit from the center of the sphere. Express the volume of $D$ as an iterated triple integral in (a) spherical, (b) cylindrical, and (c) rectangular coordinates. Then (d) find the volume by evaluating one of the three triple integrals.

$=\frac{5 \pi}{3}$

Calculus 3

Chapter 15

Multiple Integrals

Section 7

Triple Integrals in Cylindrical and Spherical Coordinates

Missouri State University

Oregon State University

Harvey Mudd College

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.

05:00

Let $D$ be the smaller cap…

13:01

Set up triple integrals fo…

11:08

17:33

Let $D$ be the region in t…

So we have room. You have the full only of our yes, fear over used to Ah, something like that. Like the like, a sphere. Whatever. Your stool. No, this shape. And we cut this fear by a plane that he's, uh, one year from the center. So this land is one. We cut it for a plane. Have you seen cup? So you obtain origin like these when we do want to compute. What is that volume? So, um, we got zero dizziness. Very cool, according it's on. So is he. You know this. Ah, Gangel feta. Cool. So all the way around? That's no problem. So there goes from zero. Wanted to play. Uh, there. So But there are restrictions for we consider Transbrasil Cut. What we're doing or here I'm saying, is this Jax is the eggs the way we consider see on? Uh, no, the Oh, no. There are. It would be just like the descends over they are. That is in cylindrical coordinates. Use very cool in cylindrical X squared plus y scored. Because you're these. Uh huh. Distance off Gruber to have ah, the boundary of the sphere and then never distance of one would be doing this region. I mean, rotating these vision over there. Eso for these regions. Uh, you can see that this angle there that he's, uh so should be such that you have that length there. Do. So you have to on in these systems, there would be from two of them school sign off that angle that should be able to One said at that angle is such a pickle. Sinus 1/2. So that the sango, uh, one you can do you from the drawing? The song should people do by thirds? Um, so that uncle there is by thirds so that, uh, our Feagles from zero off by thirds by thirds and then you can see there the restriction for row will row has to be smaller than two. And also, another restriction is that the sea coordinate Well, for doing it like this has to be bigger than one. So if we do this condition or the C chord in, it is gonna be able to row course. Well, we so that, uh, with this condition translate into one has to be No. I was more than locals. He one over, cause fee has to be one of them role on this issue, going to seek them seek until far, bro. Okay, so that, um, all the conditions for road would be that he's, uh, smarter than Sequent Sophie on babies. It is bigger than Gadhafi. And he's smaller than, uh to it's more than two for decent burial. For the volume, the volume actually equal to go there goes from Syria to buy. There is no problem. Fee goes from Sue do by thirds Seo 2/3. These with the sea on a roll goes from she isn't coffee too. To what element is rolled? Sign. So this is the volume. It's very cool. It's very goal, Koreans. No. Um, no. Do you want to do it in a cylindrical from we can You can notice that, uh, we have this region here. Yeah, if you're see, being a are simply drawing. They're evolve on dinner. Argos. Siegel's visiting one or two. That is arrange when we have these This cup there. This piece, Onda, Uh, the volume is obtained by rotating these our own theater so that, uh Well, um, you cylindrical there goes from zero off by the same on Dina. We'll see is gonna go from Warren up to two. Ah, one to usually. You see Andi in our he's gonna be constrained to be in cereal. On that He's curve there. That is currently described by saying that C square plus R squared is she going to for so that, um, we'll have that are goes off before it's called the four Miners Xi squared. So the real d c the are the bones for our would be you are zero I to square for my c squared from Dina. All the volume element ing cylindrical coordinates is our cities should be the volume. Dana, you're going to compute that in Cartesian coordinates. Well, um, but we can not. Is that, uh, ce goes from? We're gonna have this part here. Oh, okay. I wanted to So, computing these daddy see, was one of the crew sold. Um, she would go from one up to two. That should be this year on dinner. Having I haven't seen Oh, yeah. See, um, some length there. See? You have the condition that are or see square plus R squared off to these more than porter. I'm that so. These These are physical too x square. Plus. Why squared So that, um we have Ah, we could say our Why has to be, um where? Well, our excess squirt are. Why Square hostel? He's one of them. She's for moving that order. So for minus X square. Miners c squared toe. See why? What should be between minuses Screwed the for wine or six is square my nose He squared I'm screwed for my No. 60 square My us c squared on Dean R x once integrated, do you? Why? Rx square should be morning and for minus xi squared. So that explore beauty minus squared off four minus c squared. I'm screwed off. Four miners c squared. They have that in general. Um, perhaps the user integral to compute Use these one. Well, uh, so we can start be remaining first. You're going to be fair because nothing that advance on that also been drilled from Syria to buy. You said that that would be just fera began to buy zero, which is to buy So these into will turn you to he's border. He's to buy times change from one of the two firm control from one ah, cereal Up to square for minus C squared. We are we are you so that he's, um, dangerous. Look, you have lived. It's unusual coordinates. And then when we did great r D r. He's one hasn't here at the our square hubs. So that volunteer eating screw the four minor c squared zero will be or just that square house so bad a square makes this query to go away and the enough for minor C squared hunks. So they should be in each other with respect wire. So that you have to buy is William should be able to do by those. Well, well, being Joe, that was angel from one warm not to off for minus c squared from then house two councils. And we get that, people that on the irritable well, I get by dice, dear people forward before Z on the anti right above mine O c square you b minus si que third's and said that the of opportunity to one so that be called to by thanks, uh, or times two whiners through que third's miners, Uh, this week one so four times one just for minors. 1/3 eso that these would be go to these are minus four. These a plus 1/3 on these would be 8/3 minus 8/3 marries. So you have dead hands by which should be ableto okay. Minus four degrees and four on, uh, so minus. Meet my wife is, like, seven. So, with equal to that, um, he's can make a common factor. There's of the multiplied by 300 words by three. So you have that 12 here. So so minus seven. He's equal toe, so we'll be five. So you should be able to death. So five by third's should be the volume of these or work up should be able to fry well if I by thirds.

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