Approximate the area of the following using the indicated Riemann Sum.

Question

g(x) = x2 on [0,2] using 6 rectangles and using midpoint Riemann Sum.

Answer 1

You know the exact answer: x^3/3 = 2^3/3 = 8/3 = 2 2/3. Use that as a check on what you find with 6 rectangles Draw a triangle with vertices (2,4), (2,0) and (0,0) it's an upper bound area = 4

From 0 to 2 divided by 6 means the base of each rectangle is 2/6 = 1/3

the midpoints of each base are: 1/6, 3/6=1/2, 5/6, 7/6, 9/6=3/2, 11/6

1/6, 3/6, 5/6, 7/6, 9/6 and 11/6

corresponding heights of each rectangle are the squares

1/36, 9/36, 25/36, 49/36, 81/36 and 121/36

add up the heights, then multiply them by 1/3, the length of each rectangle's base

they sum to 286/36 = 143/18

(1/3)(143/18) =143/54 = 3.648

exact answer is 3.666...

It's within 0.02, a very close approximation.

do more than 6 rectangles for even closer estimates.