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

Question

what is the area and how do you find it

Answer 1

x=x^3

x^3-x =0

x(x^2-1) = 0

x=0, x= 1

area between the curves are between x=0 and x=1

integrate x-x^3 to get x^2/2- x^4/4 evaluated at x=1

1/2-1/4 = 1/4 = the area between the curves

if you estimate it with rectangles, a check is whether it's close to 1/4 or not.

if n=4 divide 1 by 4 = 1/4, you have 4 rectangular areas with bases 1/4 and midpoints of the bases = 1/8, 3/8, 5/8 and 7/8 and heights (1/8)-(1/8)^3, (3/8)-(3/8)^3, (5,8)-(5/8)^3 and (7/8)-(7/8)^3 = 63/512, 165/512, 195/512, and 105/512

multiply each base by each corresponding height

1/4 x (63/51/2 + 165/512 + 195/512 + 105/ 512)

= 1/4(512)( 63+ 165 + 195 + 105)= 528/4(512) = 132/512 = 0.2578125 which is very close to 1/4 = 0.2500000