Definite triple integral

Asked Jul 11 '20 at 09:12

Active Jul 11 '20 at 09:21

Viewed 74 times

I need to evaluate the following triple integral

$$ \int_{-n}^n \int_0^{\sqrt{n^2-x^2}} \int_0^{\sqrt{n^2-x^2 -y^2}} \; e^{-(x^2 + y^2 + z^2)^{\frac{3}{2}}}\; dz dydx $$

I converted this into the spherical coordinates. And the limits are as following

$$ 0 \leqslant r \leqslant n \\ 0 \leqslant \theta \leqslant \frac{\pi}{2} \\ 0 \leqslant \phi \leqslant \pi $$

So, the integral becomes

$$ \int_0^n \int_0^{\frac{\pi}{2}} \int_0^{\pi} e^{-n^3} \, r^2 dr \sin(\theta) d\theta d\phi $$

And after evaluating, I got the answer as

$$ \frac{\pi}{3}\, n^3 e^{-n^3} $$

Is this correct ? I tried to use Wolfram Alpha's triple integral calculator here but it could not do it. So, please check my answer.

edited Jul 11 '20 at 09:21

Gabrielek

asked Jul 11 '20 at 09:12

user9026

Why are you using LARGE font? – Gabrielek Jul 11 '20 at 09:14
To make it look nice :) – user9026 Jul 11 '20 at 09:16
2

It should be $e^{-r^3}$ not $e^{-n^3}$. – Peter Foreman Jul 11 '20 at 09:21
2

Please, don't do it again.Tex was made to be nice as it is, and there are standards that people are usual with. Everything that is out from that standards is usually seen as "non professional" – Gabrielek Jul 11 '20 at 09:23
@PeterForeman has pointed out the problem in your computation. In case you want it in wolfram alpha: link – secavara Jul 11 '20 at 09:26
Thanks Peter. And will follow the guidelines. But small font is difficult to see. – user9026 Jul 11 '20 at 09:28
But how would I enter the original integral in the Wolfram Alpha and evaluate it ? – user9026 Jul 11 '20 at 09:39

0 Answers0