Help really needed. Thank you so much

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

Given

n = 4

y₁ = x

y₂ = x³

Step One: Point(s) of Intersection

For two equations to intersect the must be equal, therefore:

x = x³

1 = x²

x = +/- 1 (For the problem we will use an upper bound b of 1)

Step Two: Determine the Top and Bottom Equations

Between (0, 1)

y₁ = x > y₂ = x³

Therefore the top equation is y₁ = x

Step Three: Determine ΔX

Δx = (b - a) / n = (1 - 0) / 4 = 1/4

Which gives the points: x₀ = 0, x₁ = 1/4, x₂ = 2/4 = 1/2, x₃ = 3/4, x₄ = 1

Step Four: Evaluate the Area of the Top and Bottom Equations

A = 1/n * sum{f((x_i + x_i-1)/2)}

A = 1/4 * (f( (x₁ + x₀)/2 ) + f( (x₂ + x₁)/2 ) + f( (x₃ + x₂)/2 ) + f( (x₄ + x₃)/2 ))

Top Equation: f(x) = x

A = 1/4 * [ 1/2(1/4 + 0) + 1/2(1/2 + 1/4) + 1/2(3/4 + 1/2) + 1/2(1 + 3/4) ]

Factor out the 1/2

A = 1/8 * [ 1/4 + 0 + 1/2 + 1/4 + 3/4 + 1/2 + 1 + 3/4 ]

A = 1/2 = 0.5

Bottom Equation: f(x) = x³

A = 1/4 * [ (1/2(1/4 + 0))³ + (1/2(1/2 + 1/4))³ + (1/2(3/4 + 1/2))³ + (1/2(1 + 3/4))³ ]

A = 1/4 * [ (1/8)³ + (3/8)³ + (5/8)³ + (7/8)³ ]

A = 31/128 ≈ 0.2422

Step Five: Area Between the Curves

A = Top - Bottom

A = 1/2 - 31/128 = 33/128 ≈ 0.2578