Applet: Double integral Riemann sum
The volume of the small boxes illustrates a Riemann sum approximating the volume under the graph of $z=f(x,y)$ over the region $D$, i.e., the double integral $\iint_D f\,dA$ for $f(x,y)=\cos^2 x + \sin^2 y$ and $D$ defined by $0 \le x \le 2$ and $0 \le y \le 1$. The volume of the boxes is $$\sum_{i,j} f(x_{i},y_{j})\Delta x \Delta y$$ where $x_i$ is the midpoint of the $i$th interval along the $x$-axis and $y_j$ is the midpoint of the $j$th interval along the $y$-axis. Drag the red and green points of the sliders to change $\Delta x$ and $\Delta y$ as well as the number of intervals along each axis. The purple line of the cyan slider shows the volume estimated by the volume of the boxes, and the blue line of the cyan slider shows the actual volume underneath the surface. As $\Delta x$ and $\Delta y$ approach zero, the purple line approaches the blue line, illustrating how the estimated volume approaches the actual volume.
Applet file: doubleriemann.m
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