Double integrals approximated with a Riemann sum

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Approximate the double integral

$\iint_R 4\,x\,y\,\dd(x,y)$ with

$R=[0,1]\times[0,1]$ via a Riemann sum over evenly divided partition with mesh size $\frac{1}{2}$ . That is, we divide the square $R$ into four sub-squares with sides of length $\frac{1}{2}$ and we index them as shown in the figure below.

Verdeling

Within each sub-square $R_{ij}$ we take the hoekpunt linksboven as tag $s_{ij}$ . Use this to calculate the Riemann sum and compare it to the exact result which is equal to $1$ .

Do the same, but now choose the hoekpunt linksonder as tag $s_{ij}$ of the subs-square $R_{ij}$

$\iint_R 4\,x\,y\,\dd(x,y)\approx{}$

with hoekpunt linksboven as tag.

$\iint_R 4\,x\,y\,\dd(x,y)\approx{}$

with hoekpunt linksonder as tag.

Multiple integrals: Double integrals

Double integrals approximated with a Riemann sum