What is incorrect with my estimation for examining the aberration regulation?

Below is a trouble in Griffiths Introduction to Electrodynamics as adheres to.

Examine the aberration theory for the function $\mathbf{v} = r^2\mathbf{\hat{r}}$, making use of as your quantity, the round of distance R, focused at the beginning?

Below is my estimation for the surface area intergral component, please aid me locating the mistake in it.

$$\mathbf{v} = r^2(\sin\theta \cos\phi \ \mathbf{\hat{x}} + \sin\theta \sin\phi \ \mathbf{\hat{y}} + \cos \theta \ \mathbf{\hat{z}})$$

\begin{align} \oint_s \mathbf{v} \cdot d\mathbf{a} &= 2 \iint r^2 \cos\theta\ dxdy \\
&= 2 \iint r^2 \cdot \frac{z}{r} dxdy \\
&= 2R \iint \sqrt{R^2 - x^2 - y^2} dxdy \\
&= 2R \int_{r=0}^{R}\sqrt{R^2 - r^2} rdr \int_{\theta=0}^{2\pi} d\theta \\
&= 4\pi R\int_{\theta=0}^{\frac{\pi}{2}}\sqrt{R^2 - R^2{\sin^2\theta}} R\sin\theta\ R\cos\theta\ d\theta \\
&= \frac{4\pi R^4}{3} \end{align}

But the appropriate solution need to be $4\pi R^4$ taking into consideration the left component of the aberration theory $\int_V \nabla\cdot\mathbf{v}\ d\tau$. I have examine my solution serveral times yet could not locate mistake, could you aid me?

many thanks.

2019-12-02 03:10:54
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