We will look at a way to find the area under a curve. One way to do this is to divide the area into rectangles. You can investigate different functions and move the slider for n to change the number of rectangles. We will look at two ways of getting our rectangles, upper rectangles and lower rectangles. You can move points A and B along the x-axis to change the boundaries for the interval.

What happens to the value of the upper and lower sums as you increase the number of rectangles? As the number of rectangles increases, does the estimate get closer to the actual value? As the number of rectangles for the Uppersum increases, the area estimate decreases. Will it continue to decrease as the number of rectangles goes towards infinity? Is there another way you could get the heights of rectangles other than Upper or Lower sums? Are there any other shapes you could divide the area under the curve into?

Explore the area under the line and the area under the quadratic. Then look at both at the same time to see how to find the area between two curves.

Will this technique work to find the area between any two curves? What if one of the curves is below the x-axis, will this technique still work? How will you find the limits of integration for the two functions?