Let $ A $ be the area under the graph of an increasing continuous function $ f $ from $ a $ to $ b $, and let $ L_n $ and $ R_n $ be the approximations to $ A $ with $ n $ subintervals using left and right endpoints, respectively.

(a) How are $ A $, $ L_n $, and $ R_n $ related?

(b) Show that $$ R_n - L_n = \frac{b - a}{n} [f(b) - f(a)] $$
Then draw a diagram to illustrate this equation by showing that the $ n $ rectangles representing $ R_n - L_n $ can be reassembled to form a single rectangle whose area is the right side of the equation.

(c) Deduce that $$ R_n - A < \frac{b - a}{n} [f(b) - f(a)] $$

If $ A $ is the area under the curve $ y = e^x $ from 1 to 3, use Exercise 27 to find a value of $ n $ such that $ R_n - A < 0.0001 $.

(a) Express the area under the curve $ y = x^5 $ from 0 to 2 as a limit.

(b) Use a computer algebra system to find the sum in your expression from part (a).

(c) Evaluate the limit in part (a).

Find the exact area of the region under the graph of $ y = e^{-x} $ from 0 to 2 by using a computer algebra system to evaluate the sum and then the limit in Example 3(a). Compare your answer with the estimate obtained in Example 3(b).

Find the exact area under the cosine curve $ y = \cos x $ from $ x = 0 $ to $ x = b $, where $ 0 \le b \le \pi/2 $. (Use a computer algebra system both to evaluate the sum and compute the limit.) In particular, what is the area if $ b = \pi/2 $?