Integral Question

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

Δx = (21 - 3)/3 = 6

L₃=(3³ + 9³ + 15³)·6 = 24786

R₃ = (9³ + 15³ + 21³)·6 = 80190

M₃ = (6³ + 12³ + 18³)·6 = 46656

T₃ = [ 3³ + 2(9³ + 15³) + 21³]·6/2 = 52488∫

∫₃²¹x³dx = x⁴/4₃²¹ = (21⁴ - 3⁴)/4 = 48600