Use the following table of values to estimate the area under the graph of f(x) over [0, 1] by computing the average of R5 and L5.
Table:
x: 0 0.2 0.4 0.6 0.8 1
f(x): 51 48 46 43 41 38
I need help with this problem.
To estimate the area under the graph of f(x) over the interval [0, 1] using the average of R5 (right endpoints) and L5 (left endpoints), we can use the formula for the midpoint Riemann sum:
R5 = (1-0) * (f(0.2) + f(0.4) + f(0.6) + f(0.8) + f(1))
= 0.2 * (48 + 46 + 43 + 41 + 38)
= 0.2 * 216
= 43.2
L5 = (1-0) * (f(0) + f(0.2) + f(0.4) + f(0.6) + f(0.8))
= 0.2 * (51 + 48 + 46 + 43 + 41)
= 0.2 * 229
= 45.8
Then, we can find the average of R5 and L5 by taking their arithmetic mean:
Average = (R5 + L5) / 2
= (43.2 + 45.8) / 2
= 89 / 2
= 44.5
Therefore, the estimated area under the graph of f(x) over the interval [0, 1] by computing the average of R5 and L5 is approximately 44.5.