The formula for the Midpoint Rule is:
∫abf(x)dx≈ Δx [ f ((x0+x1)/2) + f ((x1+x2)/2) +⋯+ f((xn−2+xn−1)/2)+ f ((xn−1+xn)/2) ]
Where Δx = (b - a)/n
In your case, f(x) = x3, a = -1.5, b = 4.5 and n = 6.
Δx = (4.5 - (-1.5))/6 = 1.
x0 = -1.5 = a
x1 = -0.5, f ((x0+x1)/2) = f(-1) = -1
x2 = 0.5, f ((x1+x2)/2) = f(0) = 0
x3 = 1.5, f ((x2+x3)/2) = f(1) = 1
x4 = 2.5, f ((x3+x4)/2) = f(2) = 8
x5 = 3.5, f ((x4+x5)/2) = f(3) = 27
x6 = 4.5 = b, f ((x5+x6)/2) = f(4) = 64
Add it alll up and then multiply by Δx = 1.
∫-1.54.5 x3 dx ≈ 99
