Let A represent the area under the graph of y = x³ between x = 0 amd x = 1. In this problem, we will follow the process in Exercise 30 to approximate A.
(a) As in (a)–(d) in Exercise 30, separately divide [0, 1] into 2, 3, 5, and 10 equal-width subintervals, and in each case compute an overestimate of A using rectangles on each subinterval whose height is the value of x³ at the right end of the subinterval.

In this case, it can be shown that if we use n equal-width subintervals, then the total area A(n) of the n rectangles is

(b) Compute A(n) for n = 2, 3, 5, 10 to verify your results from (a).
(c) Compute A(n) for n = 100, 1000, and 10,000. Use your results to conjecture what the area A equals.

Data From Exercise 30

Figure 6(A) shows two rectangles whose combined area is an overestimate of the area A under the graph of y = x² from x = 0 to x = 1. Compute the combined area of the rectangles.

We can improve the estimate by using three rectangles obtained by dividing [0, 1] into thirds, as shown in Figure 6(B). Compute the combined areas of the three rectangles.

Transcribed Image Text:

A(n) = (n + 1)² 4n²