### Estimating Double Integrals

Double integrals measure volume, and are defined as limits of double Riemann Sums. We can estimate them by forgetting about the limit, and just looking at a Riemann sum; essentially this means we're adding up the volume of boxes that fit "under" the surface *z=f(x,y)*.

The word "under" is in quotes because the boxes aren't always under the surface. We start with a rectangle in the *xy*-plane, and then draw a box over it. (Or below, if *f(x,y)* is negative there!) To decide the height of the box, we evaluate *f(x,y)* somewhere in the rectangle. Common choices include the lower-left corner of the rectangle, or the midpoint of the rectangle. In the examples below you can compare these choices for a certain function.

The pictures below show visually how you estimate the double integral of

*f(x,y)=sin*^{2}x + cos^{2}y
on the rectangle *R = [0,1] x [0, *π*]*. The three estimates shown are:

- 16 boxes, with the height evaluated at a corner of each rectangle
- 16 boxes, with the height evaluated at the midpoint of each rectangle
- 200 boxes, with the height evaluated at the midpoint of each rectangle

The actual value of the integral is π*( 1 - sin(2)/4 )*, or about 2.42743.
You can click on the pictures and rotate them; "shift"+(click and drag up/down) will zoom in and out. Press the "Home" key to reset the images.

| 16 boxes, with the height evaluated at a corner of each rectangle
Estimated Value: 2.16431
Actual Value: about 2.42743 |

| 16 boxes, with the height evaluated at the midpoint of each rectangle
Estimated Value: 2.41994
Actual Value: about 2.42743 |

| 200 boxes, with the height evaluated at the midpoint of each rectangle
Estimated Value: 2.42713
Actual Value: about 2.42743 |

Created using Live
Graphics 3D.

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