(A) Estimate the area under the graph of f(x)=52−x2 from x=−4 to x=2 using three approximating rectangles and right endpoints.

Question

(B) Repeat part (A) using left endpoints.

(C) Repeat part (A) using midpoints.

Answer 1

I will use the notation [a, b] to denote an interval on the x-axis between x = a to x = b.

The first thing we will do is divide the interval in question, [-4, 2], into three intervals of equal length: [-4, -2], [-2, 0], and [0, 2]. Next, for Part A, we plug the right endpoint of each interval into f(x). Finally, we sum the values f(-2) + f(0) + f(2) to get the appropriate Riemann approximation.

For Part B, we repeat the same procedure with the left endpoints of each interval ( f(-4) + f(-2) + f(0) ).

For Part C we repeat the procedure for the midpoint of each interval. This is the average of the two endpoints. ( f(-3) + f(-1) + f(1) ).