we use equal to n. So the width of each rectangle is . The height of the first inscribed rectangle is y 0 ,
the height of the second rectangle is y 1 , the height of the third rectangle is y 2 , and so on, up to the last
rectangle, which is yn (^) − 1. If we use the left endpoint of each rectangle, the area under the curve is
[y 0 + y 1 + y 2 + y 3 ...+ yn (^) −1]
If we use the right endpoint of each rectangle, then the formula is
[y 1 + y 2 + y 3 ...+ yn]
Now for the fun part. Remember how we said that we could make the approximation better by making
more, thinner rectangles? By letting n approach infinity, we create an infinite number of rectangles that are
infinitesimally thin. The formula for “left-endpoint” rectangles becomes
[y 0 + y 1 + y 2 + y 3 ...+ yn (^) −1]
For “right-endpoint” rectangles, the formula becomes
[y 1 + y 2 + y 3 ...+ yn]
Note: Sometimes these are called “inscribed” and
“circumscribed” rectangles, but that restricts the use of the
formula. It’s more exact to evaluate these rectangles using
right and left endpoints.
We could also find the area using the midpoint of each interval. Let’s once again use the above example,