Use the function from x = 0 to x = 1 and n equal subintervals with the function evaluated at the left-hand endpoints of each subinterval. Find a formula for the sum of the areas of the n rectangles (call this S).

Question

Use the function  from x = 0 to x = 1 and n equal subintervals with the function evaluated at the left-hand endpoints of each subinterval. Find a formula for the sum of the areas of the n rectangles (call this S). ​

Quizplus · Accepted Answer

The formula for the sum of the areas of n rectangles evaluated at the left-hand endpoints of each subinterval is:
S = Δx(f(x0) + f(x1) + f(x2) + ... + f(xn-1))
where Δx = (1-0)/n = 1/n, and xi = iΔx for i = 0, 1, 2, ..., n-1.
Substituting the given function and simplifying:
S = Δx[f(0) + f(Δx) + f(2Δx) + ... + f((n-1)Δx)]
S = (1/n)[1/(1+0) + 1/(1+Δx) + 1/(1+2Δx) + ... + 1/(1+(n-1)Δx)]
Using the formula for the sum of a finite geometric series:
S = (1/n)[1/(1-1/n)(1+0)]
S = (1/n)[n/(n-1)]
Simplify:
S = 1/(n-1)
which matches choice B.

Use the Function from X = 0 to X