double intergral, multi variable \[\[\int\limits\limits_{-r}^{r}\int\limits\limits_{-\sqrt{r^2-y^2}}^{\sqrt{r^2-y^2}}\ \sqrt{r^2-y^2} dx dy\]

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double intergral, multi variable \[\[\int\limits\limits_{-r}^{r}\int\limits\limits_{-\sqrt{r^2-y^2}}^{\sqrt{r^2-y^2}}\ \sqrt{r^2-y^2} dx dy\]

Mathematics
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we are to assume r is a constant correct?
yes
Find the volume of the solid bounded by the cylinders x^2+y^2=r^2 and y^2+z^2=r^2, and That is how i formed the intergral, if that helps anybody

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http://www.youtube.com/watch?v=k7ND70gFTLU&feature=related
http://www.youtube.com/watch?v=9FulyvBhdkw&feature=related
that should help. You can use the polar coordinates too.
the thing is for this section we haven't learned the polar coordinates, it would make this problem somewhat easier i feel
well, then, just go with the first video. That should help you evaluate multi variable double integrals.
okay thank you very much
also, he explains how to calculate double integrals for regions bounded by two curves. Keep that in mind and see if you have done it that way to evaluate your problem.
I would think that since you are calculating volume, you should have a triple integral.
http://www.youtube.com/watch?v=AQHcZklcltI That video and the part 2 of it explain how to calculate volumes bounded by two regions.
OK, if you are not using polar coordinates you have to get rid of r. \[r ^{2}=x ^{2}+y ^{2}\] Convert in your original eq see if you have something to work with.
Scratch that. I've got it. \[x ^{2}+y ^{2}=r ^{2} \] \[x ^{2}+y ^{2}=1\] That's one surface without r.
Other surface \[y ^{2}+z ^{2}=r ^{2}\] \[y ^{2}+z ^{2}=1\] \[z ^{2}=1-y ^{2}\] \[z =\sqrt{1-y ^{2}}\]
Other surface \[y ^{2}+z ^{2}=r ^{2}\] \[y ^{2}+z ^{2}=1\] \[z ^{2}=1-y ^{2}\] \[z =\sqrt{1-y ^{2}}\]
chaguanas, why are you assuming r = 1?
It is equal to \[x ^{2}+y ^{2}=1\]And I can also derive it by converting one of the equations to \[r ^{2}\cos ^{2}\theta + r ^{2}\sin ^{2}\theta = r ^{2} \] Factor the r out gives me equivalent r=1
Your region is the top semi circle of 1 which makes your x go from -1 to +1; your y goes from 0 to square root of (1-y^2)

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