Solving 3-D Dirac Delta Function Homework Question?

[Dopplershift]

Homework Statement

\begin{equation}
\int_V (r^2 - \vec{2r} \cdot \vec{r}') \ \delta^3(\vec{r} - \vec{r}') d\tau
\end{equation}

where:
\begin{equation}
\vec{r}' = 3\hat{x} + 2\hat{y} + \hat{z}
\end{equation}

Where d $\tau$ is the volume element, and V is a solid sphere with radius 4, centered at the origin.

Homework Equations

The Attempt at a Solution

I know the following:

Suppose:
\begin{equation}
\int_V f(r) \delta^3(\vec{r}-\vec{r}') d\tau = f(\vec{r'})
\end{equation}
(if r' is in the volume).

I'm just confused on how to plug in r' into f(r) which is
\begin{equation}
r^2 - 2\vec{r} \cdot \vec{r}'
\end{equation}

Any help to get me started will be much appreciated.

 

[strangerep]

$$\int_V (r^2 - \vec{2r} \cdot \vec{r}') \ \delta^3(\vec{r} - \vec{r}') d\tau $$where:
$$\vec{r}' = 3\hat{x} + 2\hat{y} + \hat{z}$$ where $d\tau$ is the volume element, and V is a solid sphere with radius 4, centered at the origin.

First, write out your integral as an explicit triple integral.

 

[Dopplershift]

First, write out your integral as an explicit triple integral.

Okay, so I can setup the integral.

\begin{equation}
\int_v (r^2 - 2\vec{r} \cdot \vec{r}') \delta_x(x-x_0) \delta_y (y-y_0) \delta_z (z-z_0) dx dy dz
\end{equation}

I guess I'm confused how I plug $\vec{r}'$ into f($\vec{r}$)

 

[Dopplershift]

Never Mind, I figured it out. I was overthinking the problem. Thanks for your help! :)