Find an estimate of the area under the graph of y= e^(sin x) between x= 0 and x= pi. Use two midpoint rectangles.

Expert Answers
lfryerda eNotes educator| Certified Educator

To find the estimate of the area under the curve using midpoint rectangles, we need to find the coordinates of the rectangles, then find the area of each rectangle, then add the areas.

Each rectangle will have a width of `pi/2` since there are two rectangles and our domain goes from `x=0` to `x=pi`.  The height of each rectangle is evaluated at the midpoint of each rectangle.  In the first, it is evaluated at `x=pi/4` and for the second, it is evaluated at `x={3pi}/4`.

This means the rectangles have coordinates

`(0,0)` to `(pi/2,0)` to `(pi/2, e^{sin(pi/4)})` to `(0, e^{sin(pi/4)})`

`(pi/2,0)` to `(pi,0)` to `(pi, e^{sin({3pi}/4)})` to `(pi/2, e^{sin({3pi/4})})`

Since `sin(pi/4)=1/{sqrt 2}` the area of the first rectangle is `pi/2 e^{-sqrt 2}`.

Since `sin({3pi}/4)=1/{sqrt 2}` the area of the second rectangle is also `pi/2 e^{-sqrt 2}` .

The estimate of the area under the graph is `pi e^{-sqrt 2}`.