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

Question

g(x) = 4x on [0,1] using 5 rectangles and using the midpoint Riemann Sum.

Answer 1

base of each rectangle is 1/5

multiply that by the sum of the heights of each rectangle

heights are the midpoints x 4 or

1/10, 3/10, 5/10, 7/10 and 9/10 each times 4

heights are

4/10, 12/10, 20/10, 28/10 and 36/10

sum is: 100/10 =10

multiply by 1/5 to get 2 which is an exact "approximation"

area is that of a right triangle with base 1 and height 4 =2

or take the integral of 4x = 4x^2/2 = 2x^2 = 2(1)^2 = 2