Use the mid-point rule with n = 4 to approximate the area of the region bounded by y = x3 and y = x

Question

Use the mid-point rule with n = 4 to approximate the area of the region bounded by y = x³ and y = x

Answer 1

First establish what the midpoints actually are. If you sketch the curves, one is a cubic and the other linear. We find the intersecting points by setting the curves equal to each other so x^3-x=0 meaning that the intersections are at -1, 0, 1. The period of coverage is then 2 in length. Dividing by n=4, makes each section 1/2 in length, so the mid-points are in the middles -3/2, -1/2, 1/2, 3/2. To evaluate the area between curves, you must take the difference between the upper and lower curves at these points. Because it is symmetrical, you can just evaluate the right side and then double it. That leaves you with 2*((1/4)-(1/4)^3 + (3/4) - (3/4)^3). I come up with 1.125