triple integral/cylindrical coordinates: 1+x/(x^2+y^2)^(1/2)

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purell
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triple integral/cylindrical coordinates: 1+x/(x^2+y^2)^(1/2)

Postby purell » Thu Apr 16, 2009 1:27 am

i have to find the triple integral of 1+x/(x^2+y^2)^(1/2) dx dy dz where S is the solid bounded by the paraboloids z=x^2+y^2 and z=1-x^2-y^2.

this is as far as i have gotten:
F(r,theta,z)=(r*cos(theta),r*sin(theta),z)
abs(det(JF))=r

so now i have the triple integral of r(1+cos(theta)) dz dr dtheta.

but how do i find the limits of integration?

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Martingale
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Re: triple integral/cylindrical coordinates

Postby Martingale » Thu Apr 16, 2009 3:31 am

purell wrote:i have to find the triple integral of 1+x/(x^2+y^2)^(1/2) dx dy dz where S is the solid bounded by the paraboloids z=x^2+y^2 and z=1-x^2-y^2.

this is as far as i have gotten:
F(r,theta,z)=(r*cos(theta),r*sin(theta),z)
abs(det(JF))=r

so now i have the triple integral of r(1+cos(theta)) dz dr dtheta.

but how do i find the limits of integration?


if your integrand is

then I like your conversion to cylindrical coordinates...

we have and

so

and

these are the upper and lower bounds for so we have



where the two paraboloids meet so

is the maximum can be.

so we should be able to write the integral as



purell
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Joined: Thu Apr 16, 2009 1:22 am
Contact:

Re: triple integral/cylindrical coordinates: 1+x/(x^2+y^2)^(1/2)

Postby purell » Thu Apr 16, 2009 5:31 pm

thanks so much!


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