Estimating the area under a curve

Suppose we want to know the area of the blue region in the figure. Until now, we have figured out the area of a polygon, which is a region surrounded by straight line segments. What should we do now that the top of the region is a curved line?

One method is to choose points on the curve and then join those points by straight line segments. The total area of the regions under the line segments can be used to estimate the area of the blue region.

To see how this is done, drag the slider under the x-axis slowly to the right to produce 2,3, or more trapezoids, shaded green. Using more and more trapezoids will provide a better estimate of the blue region's area.

The total area of the trapezoids is shown at the top of the screen, as well as the area of the blue region and the percentage difference between those areas. As you use more trapezoids, observe that the error gets to be quite small -- less than 1 per cent.

Now drag the slider back to the left to show the blue region. The top of the blue region is a curve with just a few hills and valleys. Drag the slider labeled 'wiggle controller' to the right: the top curve of the blue region will have more wiggles. You will need to use more trapezoids to get a good approximation to the area under a curve with a lot of wiggles.