Question

Which represents the midpoint Reimann sum of f(x)=−2x² +8 for −2A (−1.6)f(0.8)+(−0.8)f(0.8)+(0.8)f(0.8)+(1.6)f(0.8)

B (5)[f(−1.6)+f(−0.8)+f(0)+f(0.8)+f(1.6)]
C (−1.6)f(5)+(−0.8)f(5)+(0.8)f(5)+(1.6)f(5)
D (0.8)[f(−1.6)+f(−0.8)+f(0)+f(0.8)+f(1.6)]

Answer 1

The midpoint Riemann sum can be calculated by dividing the interval into subintervals, evaluating the function at the midpoint of each subinterval, and summing up the products. The correct representation for the midpoint Riemann sum of f(x) = -2x² + 8 for -2 ≤ x ≤ 2 is (0.8)[f(−1.6) + f(−0.8) + f(0) + f(0.8) + f(1.6)].

Step-by-step explanation:

The midpoint Riemann sum of f(x) = -2x² + 8 for -2 ≤ x ≤ 2 can be calculated by dividing the interval into subintervals and taking the midpoint of each subinterval.

The width of each subinterval is (b - a) / n, where b is the upper limit, a is the lower limit, and n is the number of subintervals.

Then, evaluate the function at the midpoint of each subinterval and multiply it by the width of the subinterval. Finally, sum up all the products to get the midpoint Riemann sum.

In this case, the correct answer is (D) (0.8)[f(−1.6) + f(−0.8) + f(0) + f(0.8) + f(1.6)].