Let f(x) = 8x Use a Riemann sum with four subintervals of equal length (n = 4) to approximate the area of R (under the graph of f on the interval [0, 2]). Choose the representative points to be the left end points of the subintervals.
Repeat previous part with eight subintervals of equal length (n = 8).
Compare the approximations obtained in previous parts with the exact area (16 sq units). Do the approximations improve with larger n?
A) 14 sq units; 12 sq units; yes
B) 14 sq units; 12 sq units; no
C) 12 sq units; 14 sq units; no
D) 12 sq units; 14 sq units; yes

Question

Quizplus · Accepted Answer

When we use four subintervals, each subinterval has length 0.5, and so the representative points are 0, 0.5, 1, and 1.5. Using the left endpoints, we evaluate f at each of these points: f(0) = 0, f(0.5) = 4, f(1) = 8, and f(1.5) = 12. The Riemann sum is thus:

Δx(f(0)+f(0.5)+f(1)+f(1.5)) = 0.5(0+4+8+12) = 12 sq units.

When we use eight subintervals, each subinterval has length 0.25, and so the representative points are 0, 0.25, 0.5, 0.75, 1, 1.25, 1.5, and 1.75. Using the left endpoints, we evaluate f at each of these points: f(0) = 0, f(0.25) = 2, f(0.5) = 4, f(0.75) = 6, f(1) = 8, f(1.25) = 10, f(1.5) = 12, and f(1.75) = 14. The Riemann sum is thus:

Δx(f(0)+f(0.25)+f(0.5)+f(0.75)+f(1)+f(1.25)+f(1.5)+f(1.75)) = 0.25(0+2+4+6+8+10+12+14) = 14 sq units.

Both approximations are off from the exact area, and neither is closer than the other. However, as n increases, the approximations tend to improve (i.e. get closer to the exact area).

Let F(x) = 8x Use a Riemann Sum with Four