Question

Consider the integral:

21
∫ x^3 dx
3

-Use Left Sum, Right Sum, Midvalue Sum, and Trapezoid Rule methods to approximate this integral with n = 3

-Find the exact value of this integral

Answer 1

I will do the exact value part, I don't know how your text or your notes
define "Left Sum, Right Sum, Midvalue Sum, and Trapezoid Rule methods"

∫ x^3 dx from 3 to 21
= [(1/4) x^4] from 3 to 21
= (1/4)(21^4) - (1/4)(3^4)
= 194481/4 - 81/4
= 48600 units^2

Answer 2

As for the approximations, you have n=3, so you want three intervals, each of width (21-3)/3 = 6. That gives you
x_₀=3, x_₁=9, x_₂=15, x=_₃21

Left Sum:
2
∑ 6f(x_{_k})
k=0

Right Sum:
3
∑ 6f(x_{_k})
k=1

Midpoint Sum:
2
∑ 6f((x_{_k}+x_{_k+1})/2)
k=0

Trapezoid Sum:
2
∑ 6(f(x_{_k})+f(x_{_k+1}
))/2
k=0