Evaluating a Triple Integral in Spherical Coordinates

[RACINGINTHESTREET91]

This is a basic example, transforming triple integrals to spherical coordinates with more complex regions should have been the follow-up video.

[publicanimal]

Can we apply Fubini's theorem for any triple integral problem involving spherical coordinates? What if you are dealing with a sphere that is not centered at the origin? My professor never introduced it, so I didn't know that you could break up the integrals like that. I have an exam tomorrow and that trick would really help me out.

[x1z3c7]

I think he broke up the integral like that because if you integrate sin(theta) form 0 to 2pi you will get 0, and the volume cannot be 0

[Invertedzero]

Whats the easiest way to remember or derive spherical coordinate conversions especially where grad and grad^2 are involved? Anyone got any tips?

[soph6ia]

You are essentially finding the volume of a sphere. Isn't the volume of a sphere 4/3 * pi * r^3? When r =1, V = 4/3 pi. He gets 4/5 pi. Wonder what the problem is.

[tangentcausality]

@soph6ia I think the relation x^2+y^2+z^2=p^2 is already accounted for in the dv formula he used (p^2*sin(phi)*dp*dphi*dtheta), so in including the function, he counted it twice and ended up with r^5/5 instead of r^3/3.

[x1z3c7]

@soph6ia I think that's because it should be d phi d theta instead of d theta d phi , with the same bounds. Tell me if I am wrong...

[sdf40197]

omg oyu are great man i am froim india and here teacher literally teaches nothing..... my end sems starts day after tommorow and today i feel i am going to pass the!!1

[missbassoon1992]

you're left handed I automatically love you. and watching your video for only a minute and I already understand more that I have been missing in my calculus class, lol

[vplof]

PLEASE PLEASE make a video of changing the order of integration in triple integrals

[jkanyce]

Oh my gah!! You can split the integrals??? My life would've been so much easier had I known this earlier. =/

Thanks so much for this!!

[bronxbeckham]

Patrick you have helped me all the way from College Algebra through Calculus 3. Is your real name Clark Kent by any chance? =) Thank you so much.

[oOKasp3rOo]

very nice!! i dont get how the hell you do this...i just stare at your video getting bits and pieces and then suddenly BAM i get the whole concept...as opposed to just getting bits and pieces :D

Thanks alot you're seriously awesome!

[flamenballerG37]

what is wrong with using (4/3)(pi)( r^3) ? r =1 in this case.

so the answer you get from this method is (4/3) pi

[VolumesofRevolution]

I was wondering the same thing.

[KevoPK]

I was trying to explain this to a 15 year old. He didn't understand it. :D

Thanks for the awesome videos Patrick!

[Matmo11]

My teacher never even told us we could break up the integral like that and multiply it out........curse that woman....

[patrickJMT]

@Matmo11 only in certain cases

[Matmo11]

@patrickJMT I see...either way, I was just tested on this today, so I can forget it for now...BUT ILL BE BACK HERE WHEN FINALS COME AROUND DAMMIT. lol

[andydrew129]

@patrickJMT

[andydrew129]

When Can you break up the integral and multiply it out?

[RSKrug]

@andydrew129 When you're integrating in terms of any variable, treat the other variables like constants. So just like the (integral of 3x) = (3 * Integral of x), the same works if you take [sin(phi) * dphi * dtheta] out of the first integral (which is in terms of rho) out of that integral into the integral in terms of theta. The dtheta stays in that integral but the sin(phi)* dphi moves out into the last integral.

[pulvid]

why to multiply by p^2sinphi????????? plzzz answer

[Matmo11]

@pulvid You just have to...JUST DO IT

[lolgepwnt]

@pulvid

That term is the determinant of the Jacobian matrix for the transformation between Cartesian and Spherical coordinates.

Same thing applies when moving from Cartesian to polar, where you replace dxdy by rdrd(theta)

[jermkent]

why does phi only go from 0 to pi? i know you explained it but i'm still unsure. thanks

[tadm123]

p always starts from the origin? from 0?

[luchinnn77]

@tadm123 it depends on the compound to be analyzed. for xample: if you have a compound bounded by 4 =< X^2 + Y^2 <= 16 your p (rho) will be starting from 2 to 4, that's on plane therms, R2

if you have 4 =< X^2 + Y^2 + Z^2 <= 16 it will be the same, but for R3 space, put Z=0 and that's all. as you can see, his R3 compound to be analyzed, it contains the origin. think about the smallest value that your radio can take, to the maximum value and that's your 'p' integral limits

[tadm123]

@luchinnn77 oh thanks man

[luchinnn77]

@tadm123 you're welcome, even if i really don't know so much about this subject, i can help on what i know. Greetings.

[adrian9790]

Gracias compadre!!!! XD 

[Yotuby93]

This is awesome

[hellstorm6]

I don't often leave comments, but I had to here. Your tutorials are amazing, truly truly helpful. Bless you sir.

[manuel7666]

Hi, I have this triple integral, integral from 0 to 2pi, integral 0 to pi / 4, 1sec integral (phi) / 2 ^ (1 / 2), rho ^ 2.Without (phi) d (rho) d (phi) d (theta)

if you can help calculate the integral, the result is (2pi / 3)-5pi / 6. (2 ^ 1 / 2)

thank you very much

[lebanesewa7sh]

bro u r awesome keep it up :D

[kristaljoy20]

oh my god? what is that i can understand every single problem

[MrDontow]

Could you do an example using spherical coordinates to find the volume of a solid bounded by another solid? Say the volume outside of a cone but inside of a sphere.

[TheOncempct]

Patrick, YOU ARE AWESOME!!!

[OrgasmicOprah]

Thank you sir.

[mattattack28]

4 dislikes? WHY? All of these videos are fantastic. Helped me through AP Calc AB and BC in high school, a semester of multivariable calc in college, and right now I'm taking the calculus that comes after that (differential equations and stuff) and these videos STILL help me. We'll see what Calc 4 is like next semester hah

Thanks Patrick!

[chasemeyers]

Sir, I would like to meet you and shake your hand!

[aadeebindy]

3 girls in the same problem woooooow.

[kizalia]

haha in aus, phi is pronounced "phy like ph-eye" and it's actually the limit that determines the angle between the xy axis not theta ahh our math is backwards.

[92310CAMILLE]

can i ask a question SIr , can different calculus find the volumes and areas like the way how integral calculus does it?

thank you sir

[chihuasmex89]

So Is it ok to say that PHI is always going to be half of theta?

[gokujo2]

@chihuasmex89 Check out the video response if you want to see a visual explanation of the limits of integration for phi :)

[LioN2765]

Hi Patrick it was great tutorial

[muslim86mus]

3:43 You mention that Fubini's theorem can be applied to bust up the integral, given the integrand is all factored.

Correct me if I'm wrong, but the theorem only states that in ITERATED integrals it's allowed to CHANGE the order of integration as long as we're integrating f(x,y) over a rectangular region.

We can bust up the integral ONLY in the special case where f(x,y) can be factored as the product of a function of x only and a function of y only.

[okara83]

I'm still not sure how the limits of integration are 0 to pi for the last term, I don't get how it accounts for the whole sphere...

[muslim86mus]

@okara83

The last term is denoted by 'phi'. Ok now, all you have to do is imagine the spherical shape of the Earth.

Take the xy-plane as the EQUATOR PLANE, dividing the Earth into north & south hemispheres. So 'phi' extends from the north pole (90° N) to south pole (90° S), summing up to 180° or π.

[muslim86mus]

@okara83

On the contrary, the range for 'theta' is [0, 2π]. As we want to have complete revolution, the EQUATOR LINE requires that we travel from Meridian Greenwich both left & right... till we meet exactly 1 point (180° W / 180° E), summing up to 360°.

Hope this helps!!

[Shadoweprinz]

Somebody explain 3:10 to 3:30, why is it only pi???

oh, and my midterm is in 1 hour 2 minutes so answer fast :P

[Shadoweprinz]

@Shadoweprinz would you look at that

theres a video response answering my question!

you gotta love youtube!

[patrickJMT]

@Shadoweprinz yes, i wish more people got involved - i would be fine with video responses

[gokujo2]

@Shadoweprinz Ohh yeah, love it! Hope you noticed the vid before your midterm :P

[Shadoweprinz]

@gokujo2 haha yeah i did :D

im actually in the UC system, so Im done with vector calc now o.o (both differentiation and integration), since were on the quarter system

[gokujo2]

@Shadoweprinz Good luck man! My cousin goes to CS long beach. and I was born in CA :) So that's cool

[TheBigTicketStub]

Just got an A on my first midterm, multivariable calc, at a top college...thanks man, you are awesome!

[mariomaruf]

Why don't you put this in your Calc 3 playlist?

[chestercy]

Thank you.

[vishnukartha47]

Excelent video, after seeing this video ,problems were easy as plucking a flower,

you change mylife THANKYOU!!!!

[slimshady147258]

.... you lost me at x to p lols >.<

[RhinoGT1]

youre the man

[sherby200]

This was very helpful

[vopsman]

i love you

[iamarealpersonsrsly]

I just finished my Calc 3 class and I got an A. This was the first A I've ever received in any Calc class I've taken and its largely in part because your videos. I only wish I had started watching sooner. Thank you so much.

[patrickJMT]

@iamarealpersonsrsly glad the class turned out well for you!

[mrfemarwashere]

sme to me i get A.. thnks utube.. and all of u tht upload nice vid.. u help me so much

[therosscompton]

Could you do another one where instead  of what you are integrating being x^2+y^2+z^2 you integrate 1/(the square root of)x^2+y^2+z^2 with the sphere being x^2+y^2+z^2=25?

[tfreed20]

very helpful, thank you

[sinareth]

thanks so much, it help about psi, i don't know how to set up the limit, i understand.

you are so nice and very helpful

[omarofuae]

I keep clicking those ads for you!!

[patrickJMT]

@omarofuae you are a good soul : )

[lalwhyumad]

@patrickJMT ah lol, ill click ALL ads for you! and idle them :D

[pvabhalamja]

HI, thanks for the explanation. Just one question. Can you explain bit more about how you got the limit of phi from 0 to Pi please?

Thank you

[gokujo2]

@pvabhalamja I made a video response just for you! I didn't see how I could make a video response to this video

[franky227]

hey patrick, can you do a more difficult spherical coord problem?

[mrman41]

@Frank12387 You can hear Patrick say minus negative cosine at the part you are talking about. The opposite of a negative number is a positive number.

or mathamatically, ((-cos(pi))-(-cos(0)) = -cos(pi) + cos(0)

[dimseeeen]

more examples please

[raitey]

Im trying to apply this to a common calculation in astrophysics trying to find the standard value for Sin^3i to find the average inclination of orbit for binary star systems... Thank god for you taking time out to share your knowledge..

Now let me keep re watching this video until i can get it to fit into my application :-/

Cheers!

ck

[RJE06]

I have final exam tomorrow ^ ^, and this video helped me..

thank you 

[sdjkl452]

I saw your explanation and I liked so much, i'm brazilian n i studin' civil engineering n 4 us it's very important. Congratulations, u did a nice job

[patrickJMT]

@sdjkl452 thanks! glad u like it!

[JFdragonfly]

Thank You Thank You Thank You!!! I get it now! =]

[Knightmetal]

Excellent video! there's only this one part with the limits of integration with Phi, I would've used from 0 to 2*Pi instead of 0 to Pi, can anybody explain that to me?

[gokujo2]

@Knightmetal Take a look at the video I made, it's on my page. I hope it helps

[stevenreid1990]

Wow man, I think you would make a great calculus prof! I can understand your way of teaching very well.

[Frank12387]

Hey Patrick, shouldn't it be (-cos(pi)-cos(0))*(2pi)*(1/5) = 0 instead? I think u made a mistake when u wrote (-cos(pi)+cos(0)) when it should be (-cos(pi)-cos(0))...

[viCoN24]

@Frank12387 -(cos(pi)-cos(0))= -cos(pi)+cos(0) ?

[mrman41]

@Frank12387 Not quite, you can hear Patrick say, minus a negative cosine as he makes the plus. The opposite of a negative number is a positive number.

[mrman41]

@Frank12387 You can hear Patrick say minus negative cosine at the part you are talking about. The opposite of a negative number is a positive number.

or mathamatically, ((-cos(pi))-(-cos(0)) = -cos(pi) + cos(0)

[krillansavillan]

wow thats a really helpful use of fubini,

does that trick only work in spherical coordinates pat?

[6138563248]

Great video and very helpful. However, you did not say or show that rho squared sin phi was derived from the Jacobian which involves the determinant of x,y,z where...

x = ρsinØcosϑ,y = ρsinØsinϑ,and z = ρcosØ.

Thanks again for clarity.

[irfaanfr1]

Thanks!

[Djole0]

Thank you very much,please dont stop with this.. u are great !!

[libyan20081985]

3:15 if some one can answer me why dQ from 0 to bay

[ncspliffy]

cheers, we are having alot of problems where our professor requires that we refer to a whole lot of theorems and definitions. I'd love to get these straightened out. Do you have any vid about using and understanding theorems like: gauss, green, stokes, fubini, Lagrange multipliers and the relationship between del, curl, grad and div?

[TomRiverstone]

Patrick, you're amazing man! I needed this stuff for my new tutoring job, I forgot most of Calc III and Diffy Q's since I took them, you are the man!!!

by the way, do you know of any website where I could find all the calculus formulas (calc I - calc III, diffy q, etc...)

[tuberALEX]

@TomRiverstone I dont think you should be a tutor if your watching these vids.... i mean students like me watch these to help us study. Tutors should be knowledgeble and flawless like JMT is

[mood902]

Man, seriously your videos are a lifesaver for students everywhere. I am workin on my degree and I can definitely say you are helping SOOO much... I might ask them to present my degree in part to you,,, lol thanks so much

[mohsen1987]

dude that is amazing, so clear.

i just got some new information from you

thanks

[Antonshavlik]

The only thing I do not understand is why we multiply bb p^2 when switching from rectangular to spherical, espcially since if we didn't we'd get 4(pi)/3 which is the volume of a sphere, when now we get 4(pi)/5 which seems wrong?\

can anyone clarify this for me?

[Antonshavlik]

never mind, clarified in the comments

[kennyoh]

this solution is incorrect. the end volume should be 4(pi)/3. the mistake is that he has a p^2 term in the triple integral. the work is good otherwise.

[hbquanie]

thats part of the formula u fool!!

[kennyoh]

o woops i thought he was trying to calculate volume.

[hbquanie]

lol yea...i realized what you were thinking afterwards..:)

[oladapo99]

dude..thank you!!

[arexxk]

WOW totally forgot that you can break up the integrals like that.. Thank you SOOO much, saved me much grief

[Jwalkin23]

Just use this for some EMAG. I forgot all of my calc III.

Goodstuff.

[corleonecico]

you explain so well in detail i wish you was my teacher

[sulwyn]

You make it seem so easy. :)

[LetMemorexZep]

nice

[creamypouf8]

I guess being a Canadian I prefer "zed" or "zee", but whatever.  Also I cringe at hearing "pee" instead of "rho" but that might be because I'm in physics.

[daltig]

Would you explain more why the z integral is 0-pi? I didn't understand your logic.

[drifter8027]

if you went 0-2pi in the z, it would be redundant. it's tough to describe without a picture, but when you have 0-2pi for theta you make a full circle. if you made another full circle for phi you'd be overlapping. try to visualize it if you can.

[Pulpdood]

C = seee, phi = phee, Z = zee... Everything rhymes!!! Ph-eye, and Zed sounds better to me, though of course phee and zee are correct as well.

anyway...

Thanks a lot - great video!!!

[gigahitme]

awesomeness you rock!!!

[MICHAELMEMISBACL]

A good analogy for why theta goes from 0 to 2pi and phi only goes from 0 to pi is this: theta is like longitude and phi is like latitude.

[promac1212]

Thanks alot

[TroyaE117]

Well explained.

[rahmanr]

THANK YOU

[JRajashekar]

Thanks

[beerfan190123]

WOW! I wish I saw this before my midterm test! Thanks, great!

[Alibatros007]

Thanks a lot...!!!!!!

[bryanrossin]

this isn't a calculation of volume, be careful. even though the region and the function both involve x^2+y^2+z^2, they are unrelated. to calculate the volume of a region using triple integrals, integrate the function f(x,y,z)=1.

this could be used to calculate mass. in that case, the f(x,y,z)=x^2+y^2+z^2 would represent the density. in other words the density of that ball increases as you move away from the origin.

just wanted to clarify. fantastic walkthrough, nice job.

[patrickJMT]

great comment! thanks!!

[eddyc27]

@bryanrossin lol i just busted by brains for so long working out why the answer was wrong (not having read this!) and kind of came to the same conclusion as you... because by using that p^2 jacobian, you're already effectively integrating within the unit sphere? so you don't need an extra p^2 ?

[qbanguy7]

very nice explanation! Thank you

[fendereater]

i have my calculus III exam tomorrow and this is saving my ass. I haven't been to class since October so I'm learning half of the course tonight. Ha. o well. Thanks!

[brvheavyhitter20]

GREAT VIDEO!....one question...i know Fubini's theorem is only applicable when the function is 'separable', but how do u know if a function is separable?

[patrickJMT]

well, just play w/ the algebra. u need it to be the product of a fuction of x and a function of y:  f(x)g(y)

[angel4life12x3]

this really helped!

[patrickJMT]

wonderful!

[lineriderdude9999]

my brain exploded .. im only in grade 9 math and your other vids really help.

[cyclicdesign]

isn't the formula for a sphere 4/3 pi r^3 ?

Wouldn't that give 4/3 pi, not 4/5 pi?

Maybe it should be rho^2 rather than rho^4 in the integrand?

I think that gives the right answer.

So what rationale should be given for using rho^2 rather than rho^4? ie why shouldn't we use rho^4?

[tippinsouth]

we are not finding volume

[cyclicdesign]

I think it is volume. What else would you be integrating for?

[tippinsouth]

could be density... ask patrick what he wanted to find in this case

[mostythemosty1]

it is the volume inside for the sphere

[tippinsouth]

great! we are just doing this in class :)

[patrickJMT]

i know! i was there. that is why i made this video!