Calculating Divergence Using the Divergence Theorem

[fluxions]

Homework Statement

the problem is to calculate
$$\int (\nabla \cdot \vec{F}) d\tau$$
over the region
$$x^2 + y^2 + x^2 \leq 25$$
where
$$\vec{F} = (x^2 + y^2 + x^2)(x\hat{i} +y\hat{j} + z\hat{k})$$
in the simplest manner possible.

Homework Equations

divergence theorem!

The Attempt at a Solution

Write
$$\vec{F} = |\vec{r}|^2 \vec{r} = |\vec{r}|^3 \hat{r},$$
so
$$\vec{F} \cdot \hat{n} = \vec{F} \cdot \hat{r} = |\vec{r}|^3 \hat{r} \cdot \hat{r} = |\vec{r}|^3 = 125,$$
since
$$\hat{n} = \hat{r}$$
and
$$|\vec{r}| = 5$$
along the surface of the sphere.
Then, invoking the divergence theorem, we obtain:
$$\int (\nabla \cdot \vec{F}) d\tau = \oint_{\partial{\tau}} \vec{F} \cdot \hat{n} d\sigma = \oint_{\partial{\tau}} 125 d\sigma = 125 \cdot 4 \cdot \pi \cdot 5^2$$

the back of the book gives 100pi as the answer (and I've checked the errata for the book; no correction has been made). am i wrong? or is the book?

 

[Dick]

It's also not very hard to integrate the divergence over the interior of the sphere. I get 4*pi*5^5. It sure looks to me like the book answer is wrong.

 

[fluxions]

It's also not very hard to integrate the divergence over the interior of the sphere. I get 4*pi*5^5. It sure looks to me like the book answer is wrong.

Great, thanks much!