Evaluate the following integral as a limit of a sum. ∫ x3 dx , x ∈ [1,3]

Question

Evaluate the following integral as a limit of a sum.

\(\int_1^3 x^3 \,dx\)

∫ x³ dx , x ∈ [1,3]

Answer 1

Let f(x) = x³ , for 1 ≤ x ≤ 3.

Divide the closed interval [1, 3] into n equal su bintervals each of length h at the points

1, 1 + h, 1 + 2h, ……, 1 + rh, ……, 1 + nh = 3 

∴ nh = 2

∴ h = \(\cfrac{2}{n}\) and as n → ∞, h → 0

Here a = 1