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

Question

1. f(x) = x + 2 on [0,1] using 4 rectangles and usingthe left Riemann Sum.

Answer 1

left end of each rectangle base is 0, 1/4, 2/4 and 3/4

length of each base is 1/4

height of each rectangle is 8/4, 9/4, 10/4 and 11/4

sum the heights = 38/4 = 19/2

multiply by 1/4 to get 19/8 = 2.375 = area

it's a right triangle with height 3 and base 1, area= 3/2 =1.5 + a rectangle 1 x 1,=1 total area =2.5

take the integral of x+2 = x^2/2 +2x = 1/2 +2 = 2.5

2.375 is less than the exact, as any left Reimann sum is, but it's close