- #1
tnutty
- 326
- 1
Homework Statement
The actual question is to evaluate the integral. All I need help on is the setting up part.
Instead of making a thread for each, I will post 3 integral question with my attempts.
Just tell me if you see something wrong from rectangular to spherical conversion. Thanks.
1) Evaluate the integral
[tex]\int\int\int_{E} e^\sqrt(x^2+y^2+z^2) dv[/tex]
where E is enclosed by the sphere x^2+y^2+z^2 = 9 in the first octant
This is what I got in spherical coordinate :
[tex]\int^{\pi/2}_{0}[/tex][tex]\int^{\pi/2}_{0}[/tex][tex]\int^{3}_{0} e^\rho\rho^2 *d\rho*sin(\phi)d\phi*d\theta[/tex]
Now for the seconds one :
Evaluate the integral :
[tex]\int\int\int_{E}x^2 dV[/tex]
where E is bounded by the x-z plane and the hemisphere y = sqrt(9-x^2-z^2) and y = sqrt(16 - x^2 - z^2)
Here is my integral setup in spherical coordinates :
[tex]\int^{\pi}_{0}[/tex][tex]\int^{\pi/2}_{0}[/tex][tex]\int^{4}_{3} (p^2*sin^2(\phi)*cos^2(\theta) )* p^2d\phi * sin(\phi) d\phi * d\theta[/tex]
And for the last one :
Evaluate the integral by converting it into spherical coordinate :
In rectangular coordinate :
[tex]\int^{1}_{0}[/tex][tex]\int^{\sqrt(1-x^2}_{0}[/tex][tex]\int^{\sqrt(2-x^2+y^2)}_{\sqrt(x^2+y^2)} xz *dzdydx[/tex]
Here is my partial attempt to convert it into spherical coordinate, but some are missing because I am not sure what it is supposed to be.
[tex]\int^{2*pi}_{0}[/tex] [tex]\int^{?}_{?}[/tex] [tex]\int^{2}_{?} xz \rho^2 d\rho sin(\phi)d\phi d\theta[/tex]
where x = p sin(phi) cos(theta) and z = p*cos(phi)
Thanks for any help.