Question

please help my double integration

Answer 1

Question ??

Answer 2

\[\int\limits_{0}^{2 pi} \int\limits_{0}^{1-\cos \theta}y^3\cos^2\theta dy d \theta\]

Answer 3

@ shivam_bhalla can you please help me this?

Answer 4

yes sure. First integrate
\[ \int\limits\limits_{0}^{1-\cos \theta}y^3\cos^2\theta d \theta\]
In this case y is a constant, so take it out
\[ y^3\int\limits\limits_{0}^{1-\cos \theta}\cos^2\theta d \theta\]

Then the resulting expression , integrate with respect to dy keeping theta as constant

Answer 5

then whats next?

Answer 6

Did you also do the following I mentioned above :
"Then the resulting expression , integrate with respect to dy keeping theta as constant" ?

Answer 7

the cos now will become sin^2

Answer 8

@roselynECE , no

Answer 9

@shivam_bhalla I think you are doing it backwards

Answer 10

@TuringTest , does it matter ??

Answer 11

( a doubt only)

Answer 12

the bounds on y are\[0\le y\le1-\cos\theta\]yes it matters a lot
if you do it the other way you won't get a constant for an answer

Answer 13

please show me how?

Answer 14

Thanks @TuringTest . I tried to follow wikipedia. http://en.wikipedia.org/wiki/Multiple_integral#Double_integral

@roselynECE , integration of cos^2 theta is not sin^2 (theta)

Answer 15

you should write\[ \cos^2(\theta)= \frac{1+\cos 2(\theta)}{2}\] and then try to integrate

Answer 16

you work from the inside out\[\int_0^{2\pi}\left[\int_0^{1-\cos\theta}y^3dy\right]\cos^2\theta d\theta\]first do the integral in the brackets

Answer 17

the first integral is with respect to dy, so we can treat everything else as constant, as shivam said
this is going to get ugly it looks like though

Answer 18

@TuringTest , here you have to follow wopposite of FOIL, right ??

Answer 19

well it's going to be (1-cosx)^4 so you are going to have to expand that sucker with the binomial theorem or something I guess :S
then use your formula to reduce out the squares and such

Answer 20

there is probably a faster way to do it, but I can't see how offhand

Answer 21

@TuringTest , the binomial expansion is the only way.(in my opinion)
Secondly. Do you have to follow opposite of FOIL in double integrals, right ??

Answer 22

not sure what you mean by the "opposite of foil"
you mean like factoring
x^2+2x+1=(x+1)^2
??

Answer 23

@TuringTest , got the easier way

we can write 1-cos (theta) = 2 sin^2( theta/2 )

Answer 24

nice :)
I think that will pay off

Answer 25

@TuringTest , I meant that you have into integrate as inner part first and outer part later. Everytime, right ??

Answer 26

yes, unless you are integrating over a rectangle
then you can actually separate the integrals

Answer 27

@TuringTest , can you suggest a guide for this ??

Answer 28

you could always switch the bounds too, but that sometimes makes the integral impossible

Answer 29

yeah hold on, I'm looking one up

Answer 30

http://tutorial.math.lamar.edu/Classes/CalcIII/IteratedIntegrals.aspx
I always refer people to Paul's notes lol

Answer 31

Thanks a lot @TuringTest

Answer 32

no problem, happy learning @shivam_bhalla :)

Answer 33

@roselynECE if you are still stuck I suggest you just do that integral in the brackets for starters

Answer 34

how about this int_{(0)^(6)} int_{(0)^(12-2y)} int_{(0)^(4-(2/3)(y)-(x/3)} x dz dx dy

Answer 35

\[ \int_0^6\int_0^{12-2y} \int_0^{4-(2/3)(y)-(x/3)} x dz dx dy\]???

Answer 36

OMG, that we would be ultra complicated

Answer 37

ugly ugly...

Answer 38

that x in front is going to make this especially hideous
please post this question separately anyway
did I type it right?

Answer 39

i think like this: