Hello, I need help to write formally the answer to the following problem

Define a measure on $\displaystyle \mathcal{R}^2$ (the Borel subsets of $\displaystyle \mathbb{R}^2$) by

$\displaystyle \mu((a,b]\times(c,d]) = I(a < 0 \leq b)[\max\{\min(d,1)-max(c,0),0\}]$

a. let $\displaystyle f(x,y)=(1+x)^2$. Find $\displaystyle \int f d\mu$

I did a graph, and I think this should be 1

b. let$\displaystyle g(x,y)=y^2$. Find $\displaystyle \int g d\mu$

Same deal as above, I did a graph and I think it has to be 1/3

If I split

$\displaystyle \mu((a,b]\times(c,d]) = I(a < 0 \leq b)[\max\{\min(d,1)-max(c,0),0\}]$

into

$\displaystyle \mu_1((a,b])=I(a < 0 \leq b)$

and

$\displaystyle \mu_2((c,d]) = [\max\{\min(d,1)-max(c,0),0\}]$

Both $\displaystyle \mu_1$ and $\displaystyle \mu_2$ are $\displaystyle \sigma$-finite measures, also is $\displaystyle \mu_2$ the Lebesgue measure restricted to $\displaystyle (0,1]$?

Am I on the right track?

Thanks in advance,