Triple integral: $\iiint_W\ dx\ dy\ dz$ where $W=\{(x,y,z)\in\mathbb{R}^3:x^2+y^2+z^2 \le r^2\}$

Question

I am trying to solve the following triple integral using spherical coordinates:

$$\iiint_W\ dx\ dy\ dz$$ where $$W=\{(x,y,z)\in\mathbb{R}^3:x^2+y^2+z^2 \le r^2\}$$

What I've tried:

After drafting the region it appears to me that: $$ 0 \le r \le \ ? \\ 0 \le \theta \le \frac{\pi}{2} \\ 0 \le \phi \le 2\pi $$

But I am not sure how to convert the equation of the region to spherical coordinates and to determine the value of $r$.

Are my determinations correct? How do i proceed?

Answer 1

In this problem, $r$ is not a variable, but a parameter, like $\pi$, except we don't know what it is, the variable $r$, the distance from the origin, will be between $0$ and the parameter $r$, it helps if you change the parameter $r$ to $d$ for distance from origin or something else

Also note that $\theta\in[0,2\pi]$ and $\phi\in[0,\pi]$ since $\theta$ must encompass the entire circle while $\phi $ must go from axis to axis, half a circle

Answer 2

You are mixing letters. If I was you, I would have written

$$W=\left\{(x,y,z)\in\mathbb{R}^3:x^2+y^2+z^2 \le R^2\right\}$$

so

$$0\leq r\leq R$$

Also, note that you should have

$$0\leq\theta\leq\pi$$

The angles are defined in the following sketch