Question

Estimate the area under the graph of f(x)= x^2 + 3 x from x=1 to x=10 using the areas of 3 rectangles of equal width, with heights of the rectangles determined by the height of the curve at

a) left endpoints:
b) right endpoints:

Answer 1

so the values of x would be
x = 1,4,7, and 10 , equal width = 3

left end-points
x = 1 , 4, 7
f(1) = 1+3 = 4
f(4) = 16+12 = 28
f(7) = 49+21 = 70
area = 3(4) + 3(28) + 3(70) = 306 ---> under-estimation

right end-points
x = 4 , 7, and 10
f(4) = 28
f(7) = 70
f(10) = 130
area = 3(28+70+130) = 684 ---> over-estimation

let's average them:
(306+684)/2 = 495

which is close to what if we had taken the midpoint of the base of each rectangle
f(2.5) = 13.75
f(5.5) = 46.75
f(8.5) = 97.75
area = 3(13.75+46.75+97.75) = 474.75

exact answer by Calculus:
481.5