Use the Midpoint Rule with n = 4 to approximate the area of the region bounded by the graph of f and the x-axis over the interval. (Round your answer to four decimal places.)

Question

f(x) = x6 − x7

[0, 1]

Answer 1

Δx= (1 - 0)/4 = 1/4; A ≈ ∑₀³((1/8 + i/4)⁶ - (1/8 + i/4)⁷)·1/4 = .020048

For calculations Iwe used sum(seq( functionson TI-84

Answer 2

You are approximating this area by getting the area of 4 rectangles (since n = 4).

The width of the whole interval is 1 (1 - 0 = 1). So the width of each sub-interval, and therefore the width of each rectangle, = 1/4.

The height of each rectangle is determined by the y-value of the function at the midpoint of each sub-interval. The subints are 0 - 1/4 , 1/4 - 1/2 , 1/2 - 3/4 , and 3/4 - 1. So the mdpts are x = 1/8 , 3/8 , 5/8 , and 7/8.

Lastly, the total area is calculated (with a calculator) like this:

1/4 * f(1/8) + 1/4 * f(3/8) + 1/4 * f(5/8) + 1/4 * f(7/8) ~ .020048 sq. units

This area can be calculated exactly by integrating the function on the interval [0 , 1]. The area is ~ .017857