What is the total charge of a solid sphere with given charge density?

[Uku]

Homework Statement

I am given a charge density for a solid sphere
$$\rho=14.1\frac{pC}{m^{3}}\frac{r}{R}$$
The r is the distance from the center of the sphere and R is the radius of the whole thing.

$$R=5,6cm$$

Now I am asked for the whole charge contained by the sphere.

Homework Equations

$$\rho=\frac{dq}{dV}$$

The Attempt at a Solution

$$dq=\rho dV$$
$$dq=4.1\frac{pC}{m^{3}}\frac{r}{R} dV$$
I'll just denote the picocoulomb into B
$$q=\frac{B}{R} \int r dV$$

Right, here I land. This is from Halliday, second year thing, I bet they don't expect you to do volume integration in spherical coordinates or anything such. I could write it:
$$dV=\frac{4}{3} \pi dr^{3}$$?

Pff...

EDIT:

Ok, now I get it I think:
$$q=\frac{B}{R} \int r dV$$
is actually
$$q=B \int dV$$
$$q=14.1\frac{pC}{m^{3}} \frac{4}{3} \pi r^{3}$$

ought to give me the right answer

EDIT:

It does not.
The right answer is given by
$$q=14.1\frac{pC}{m^{3}} \pi r^{3}$$

But how do I land that?

 

[cupid.callin]

just consider a small spherical shell of radius x(<R) and thickness dx

write dq for it (by finding volume of "shell" --- not of sphere of radius x----) and integrate it from x=0 to x=R