Estimate the area under the graph of 𝑓 ( 𝑥) = 𝑥^2 + 5 between 𝑥 = −2 and 𝑥 = 1 using 3 rectangles and

Question

Estimate the area under the graph of 𝑓 ( 𝑥) = 𝑥^2 + 5 between 𝑥 = −2 and 𝑥 = 1 using 3 rectangles and a. Right endpoints b. Left endpoints c. Midpoints

Answer 1

first we need f(-2) = 9, f(-1) = 6, f(0) = 5 and f(1) = 6, as well as

f(-1.5) = 7.25, f(-0.5) = 5.25 and f(0.5) = 5.25

(a) right endpoints:

x=-1, 0, 1

6*1 + 5*1 + 6*1 = 17

(b) left endpoints:

9*1 + 6*1 + 6*1 = 21

(c) midpoints

7.25*1 + 5.25*1 + 5.25 *1 = 17.75

the real area is (1/3)x^3 + 5x where x = -2..1

(1/3)*1^2 + 5(1) - [(1/3)*(-2)^3 + 5(-2)]

(5 and 1/3) - [ -8/3 + -10) =

(5 and 1/3) - ( -38/3) =

(5 and 1/3) + 12 and 2/3

18