Cylindrial Coordinates to evaluate the integral

[nysnacc]

Homework Statement

Homework Equations

x^2+y^2 = r^2

The Attempt at a Solution

I think it is asking me to find the volume of the sphere, which is in the first octant (1/8 of the sphere)

So I set
0<= z<= √1-r²
0 <= r <= 1
0<=θ<=π/2

 

[Ssnow]

Hi, remember that if you want the volume of $D$ you must calculate $\int\int\int_{D}1dxdydz$ where $f(x,y,z)=1$ ...

 

[nysnacc]

I think the question wants me to use the du dv

 

[Ssnow]

I think it is asking me to find the volume of the sphere

no because $f(x,y,z)\not=1$ ...

 

[nysnacc]

?

 

[Ssnow]

What I want to say is that your coordinates are ok $x=r\cos{\theta}, y=r\sin{\theta}, z=z$ but evaluating the integral you don't have the volume of the sphere (as you said ...) but simply a result that is the evaluation of the integral ...

 

[nysnacc]

So simply put, do I just plug in the limits:
0<= z<= √1-r2
0 <= r <= 1
0<=θ<=π/2

and into the ze^x2+y2+z2

then the integral will be something like ∫∫∫∫ z dz r dr dθ

 

[LCKurtz]

So simply put, do I just plug in the limits:
0<= z<= √1-r2
0 <= r <= 1
0<=θ<=π/2

and into the ze^x2+y2+z2

then the integral will be something like ∫∫∫∫ z dz r dr dθ

Your integrand will be $ze^{x^2+y^2+z^2}$ in cylindrical coordinates with $dV = r~dz~dr~d\theta$.

 

[nysnacc]

Your integrand will be $ze^{x^2+y^2+z^2}$ in cylindrical coordinates with $dV = r~dz~dr~d\theta$.

Okay but where does z goes (it was before $e^{x^2+y^2+z^2}$)

 

[LCKurtz]

Okay but where does z goes (it was before $e^{x^2+y^2+z^2}$)

?. Right where I have it.

 

[nysnacc]

I rewrite my thought
Originally I have limits:
0<= z<= √1-r2
0 <= r <= 1
0<=θ<=π/2

Then I use them on ∫ ∫ ∫

replace also e^{x^2+y^2+z^2} with e^r2+z2

Then I have ∫ ∫ ∫ z e^r2+z2 r dz dr dθ