Find the sum of the areas of 10 circumscribed rectangles for each curve and show that the exact…

Find the sum of the areas of 10 circumscribed rectangles for each curve and show that the exact area between the sum of the areas of the circumscribed rectangles and the inscribed rectangles (as found in Exercises 5(b)–8(b)). Also, note that the mean of the two sums is close to the exact value

Exercises 5

Find the approximate area under the curves of the given equations by dividing the indicated intervals into n subintervals and then add up the areas of the inscribed rectangles. There are two values of for each exercise and therefore two approximations for each area. The height of each rectangle may be found by evaluating the function for the proper value of x. See Example 1.

EXAMPLE 1 Sum areas of inscribed rectangles

Approximate the area in the first quadrant to the left of the line x = 4 and under the Parabola y = x2 + 1. Here, “under” means between the curve and the x-axis. First, make this approximation by inscribing two rectangles of equal width under the parabola and finding the sum of the areas of these rectangles. Then, improve the approximation by repeating the process with eight inscribed rectangles. The area to be approximated is shown in Fig. 25.4(a). The area with two rectangles inscribed under the curve is shown in Fig. 25.4(b). The first approximation, admittedly small, of the area can be found by adding the areas of the two rectangles. Both rectangles have a width of 2. The left rectangle is 1 unit high, and the right rectangle is 5 units high. Thus, the area of the two rectangles is

A much better approximation is found by inscribing the eight rectangles as shown in Fig. 25.4(c).

Exercises 8

Find the approximate area under the curves of the given equations by dividing the indicated intervals into n subintervals and then add up the areas of the inscribed rectangles. There are two values of for each exercise and therefore two approximations for each area. The height of each rectangle may be found by evaluating the function for the proper value of x. See Example 1.

EXAMPLE 1 Sum areas of inscribed rectangles

Approximate the area in the first quadrant to the left of the line x = 4 and under the Parabola y = x2 + 1. Here, “under” means between the curve and the x-axis. First, make this approximation by inscribing two rectangles of equal width under the parabola and finding the sum of the areas of these rectangles. Then, improve the approximation by repeating the process with eight inscribed rectangles. The area to be approximated is shown in Fig. 25.4(a). The area with two rectangles inscribed under the curve is shown in Fig. 25.4(b). The first approximation, admittedly small, of the area can be found by adding the areas of the two rectangles. Both rectangles have a width of 2. The left rectangle is 1 unit high, and the right rectangle is 5 units high. Thus, the area of the two rectangles is

A much better approximation is found by inscribing the eight rectangles as shown in Fig. 25.4(c).