Integrate $\iiint_S e^{\sqrt{(x^2+y^2+x^2)^3}}$ using spherical coordinates

Question

I need to integrate:

$$\iiint_S e^{\sqrt{(x^2+y^2+x^2)^3}}$$

Where $S$ is a sphere with radius 1 and center in the origin, using spherical coordinates

Using: $\iiint_s f(\rho,\phi, \theta)\rho^2 sen\phi \space d\rho \space d\phi \space d\theta$

The first thing I did was calculate the limits of integration:

for $\rho$ is $[0,1]$ because the sphere has radius $1$, for $\phi$ is $[0,\pi]$, for $\theta$ is $[0,2\pi]$.

Because $\sqrt{(x^2+y^2+x^2)}=\rho$ then $e^{\sqrt{(x^2+y^2+x^2)^3}}= e^{\sqrt{(\rho)^3}}$

so the integral I got is:

$$ \int_0^{2\pi} \int_0^{\pi} \int_0^1 e^{\sqrt{(\rho)^3}} \rho^2 sen\phi \space d\rho \space d\phi \space d\theta $$

Is this the right integral? or I did something wrong?

Answer 1

It is almost fine, indeed we have

$$\int_0^{2\pi} \int_0^{\pi} \int_0^1 e^{\sqrt{(\color{red}{\rho^2})^3}} \rho^2 sen\phi \space d\rho \space d\phi \space d\theta=\int_0^{2\pi} \int_0^{\pi} \int_0^1 e^{\rho^3} \rho^2 sen\phi \space d\rho \space d\phi \space d\theta$$