When one integrates a definite integral, one is actually finding the
area between the x-axis and the curve given by the integrand.
That is, the area under the curve f(x) between a and b is
Note that one considers the area above the x-axis as positive area, while the area below the x-axis is negative area.
One may estimate this area by filling the space with simple geometric shapes (like rectangles, triangles, cirles, and trapezoids), calculating the area that the shapes fill, and adding the positive and negative areas.
This will only be an estimate of the actual area for most curves since the sides of our simple geometric shapes will not exactly fit the curve.
This is exactly what is done with Reimann Sums where one estimates the area
with rectangles of fixed width.
The integral is then the limit as the rectangle width goes to 0 of the sum of the area of
Creator: CJ Kentler