QUIZ #12: CALCULUS 1A (Stankova) Wednesday, April 21, 2004 Section 10:00–11:00 (Voight)

Problem 1. Express the limit as a definite integral on the given interval:

lim_{n→∞} ∑_{i=1}^{n} (e^{(x_{i}^{∗})^{2}} + x_{i}^{∗}\sqrt{1 − x_{i}^{∗}}) Δx, [1, 4].

Solution. This is the midpoint formula for the integral

lim_{n→∞} ∑_{i=1}^{n} f(x_{i}^{∗})Δx = ∫_{a}^{b} f(x) dx.

Therefore the limit is

∫_{1}^{4} (e^{x^{2}} + x\sqrt{1 − x}) dx.

QUIZ #12: CALCULUS 1A (Stankova) Wednesday, April 21, 2004 Section 11:00–12:00 (Voight)

Problem 1. The graph of f is shown. Draw a picture and estimate ∫_{−2}^{4} g(x) dx with three subintervals using:

(a) left endpoints, (b) midpoints.

Solution. We have the three intervals [−2, 0], [0, 2], [2, 4], each of length Δx = 2. For left endpoints, we have the values

f(−2) = 2, f(0) = 1, f(2) = 0.

Therefore the estimate of the integral is

∑_{i=1}^{3} f(x_{i})Δx = (2 + 1 + 0)(2) = 6.

For midpoints, we have

f(−1) = 3, f(1) = 0, f(3) = 2

hence

∑_{i=1}^{3} f(x_{i}^{∗})Δx = (3 + 0 + 2)(2) = 10.