Use the mid-point rule with n = 4 to approximate the area of the region bounded by y = x3 and y = x. (10 points)

See the graph attached.

The midpoint rule states that you can calculate the area under a curve by using the formula:


In your case:
a = 0
b = 1
n = 4
x₀ = 0
x₁ = 1/4
x₂ = 1/2
x₃ = 3/4
x₄ = 1

Therefore, you'll have:



Now, to evaluate your f(x), you need to look at the graph and notice that:
f(x) = x - x³

Therefore:




M₄ = 1/4 · (2 - 478/512)
     = 0.2666

Hence, the area of the region bounded by y = x³ and y = x is approximately 0.267 square units.