(Note: The fractional subscript means to evaluate the function at the number halfway between each
integral pair of values of n.)
When you take the limit of this infinite sum, you get the integral. (You knew the integral had to show up
sometime, didn’t you?) Actually, we write the integral as
(x^2 + 2) dxIt is called a definite integral, and it means that we’re finding the area under the curve x^2 + 2 from x = 1
to x = 3. (We’ll discuss how to evaluate this in a moment.) On the AP Exam, you’ll only be asked to
divide the region into a small number of rectangles, so it won’t be very hard. Let’s do an example.
Example 1: Approximate the area under the curve y = x^3 from x = 2 to x = 3 using four left-endpoint
rectangles.
Draw four rectangles that look like the following: