Find the volume of the region inside the sphere and cylinder?

[khfrekek92]

Homework Statement

Find the volume of the region inside the sphere x^2+y^2+z^2=9, under the xy-plane, and inside the cylinder x^2+y^2=5.

Homework Equations

Need to use integrals in eaither cylindrical or spherical I'm guessing?

The Attempt at a Solution

I've graphed it and I know what region I'm evaluating, but I don't know how to set up the integral.. Thanks for any help!

 

[tiny-tim]

welcome to pf!

hi khfrekek92 ! welcome to pf!

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i'd use cylindrical (rather than spherical), because then there's no tricky angle integration …

can you describe in words what the limits are on cylindrical r and z in this case?

 

[khfrekek92]

Thank you so much! :) And thanks for that useful hint too haha ;)
P.S. The question actually asks for the top portion of this sphere instead of the bottom, soooo... so I'm guessing I would do the triple integral (rdzdthetadr) with z=[0,sqrt(9-r^2)], theta=[0,2pi], and r=[0,sqrt5] ?

 

[khfrekek92]

Thank you so much! :) And thanks for that useful hint too haha ;)
P.S. The question actually asks for the top portion of this sphere instead of the bottom, soooo... so I'm guessing I would do the triple integral (rdzdthetadr) with z=[0,sqrt(9-r^2)], theta=[0,2pi], and r=[0,sqrt5] ?

 

[tiny-tim]

… I would do the triple integral (rdzdthetadr) with z=[0,sqrt(9-r^2)], theta=[0,2pi], and r=[0,sqrt5] ?

yup!

 

[khfrekek92]

Yay! Thanks so much tiny-tim! :)