Estimate the area under the graph of

Question

(A) Estimate the area under the graph of f(x)=16-x²,

from x=0 to x=4 using 4 approximating rectangles and right endpoints estimate

left endpoints estimate:

Right endpoints estimate

Answer 1

actual area = the integral of 16-x^2 evaluated from 0 to 4 = 16x-x^3/3 = 16(4)-4^3/3 = (2/3)64 = 42 2/3

do four rectangles with midpoints, you get 43

do four rectangles from the left or right, you get 34 or 50

15+12+7 = 34

16+15+12+7 = 50

or use average

(50+34)/2 = 84/2 = 42 = very close to 42 2/3

Answer 2

Begin by drawing the graphs and rectangles for the left/right endpoint estimations to assist with visualizing the area.

Left Endpoint Estimation:

L = (1)(16) + (1)(15) + (1)(12) + (1)(7)

L = 50 (Overestimation)

Right Endpoint Estimation:

R = (1)(0) + (1)(7) + (1)(12) + (1)(15)

R = 34 (Underestimation)

ANSWERS: L = 50 and R = 34