Given the function $f(x,y) = xy$, how do I calculate $$\int_{[0,1]\times [0,1]}f(x,y)$$ by the definition of the Riemann integral, without showing it is Riemann integrable, since this comes from continuity on f?
Here are the definitions to be used:
Jordan content: If $ I = [a_1,b_1]\times[a_2,b_2]\times\ldots\times[a_N,b_N]\subset\mathbb{R}^N $ is a compact interval, the its Jordan content, $\mu(I)$, is $$ \mu(I) = \prod_{j=1}^N(b_j-a_j) $$
Riemann Sum: Let $I\subset\mathbb{R}^N$ be a compact interval. Let $f:I\rightarrow\mathbb{R}^K$ be a function. Let $P$ be a partition of $I$ into $I_\nu$. For each $\nu$, choose some $x_\nu\in I_\nu$. Then the Riemann sum of $f$ corresponding to $P$ is $$ S(f,P) = \sum_{\nu}f(x_\nu)\mu(I_\nu) $$
Riemann Integral: Let $I\subset\mathbb{R}^N$ be a compact interval. Let $f:I\rightarrow\mathbb{R}^K$ be a function. Suppose that there is $y\in \mathbb{R}^N$ such that for each $\epsilon>0$, there is a partition $P_\epsilon$ of $I$ such that for each refinement $P$ of $P_\epsilon$ and for any Riemann sum $S(f,P)$ corresponding to $P$, we have $\|S(f,P)-y\|<\epsilon$. Then $f$ is said to be Riemann integrable on $I$, and $y$ is called the Riemann integral of $f$ over $I$, so $$y = \int_{I}f$$
To calculate the integral, do I need to find a specific partition of $[0,1]\times[0,1]$ into $I_\nu$, choose a specific $x_\nu$ for each $I_\nu$, then compute the Riemann sum? If so, how do I know I'm choosing a good partition?