Triple integration over portion of Sphere

[nysnacc]

Homework Statement

Homework Equations

spherical Jacobean

The Attempt at a Solution

I have (sorry, have to capture my work, too hard to type)

then the integration of p³ e^p2 = 1/2 e^p2 (p²-3/2) ??

 

[nysnacc]

The answer I got using cylindrical was eπ/8,
but here, using the spherical, I got π(e+3)/64

 

[Ray Vickson]

Homework Statement

View attachment 106917

Homework Equations

spherical Jacobean

The Attempt at a Solution

I have (sorry, have to capture my work, too hard to type)
View attachment 106918

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Your material is NOT too hard to type; you just need to learn how to use LaTeX. That does take some effort, though.
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then the integration of p³ e^p2 = 1/2 e^p2 (p²-3/2) ??

No. You can check that for yourself, by showing that the derivative is not $r^3 \exp(r^2)$.

Do integration by parts.

BTW: I get $\pi/8$ using spherical coordinates.

 

[nysnacc]

π/8 not πe/8 ?

 

[Ray Vickson]

π/8 not πe/8 ?

I wrote exactly what I meant.

If you had a different upper bound of $r = a$ instead of $r = 1$ you would, indeed, get an exponential in the answer.