Evaluate the following integrals as a limit of a sum. ∫ (3x - 4).dx , x ∈ [1,3]

Question

 Evaluate the following integral as a limit of a sum.

\(\int_1^3 (3x-4).dx\)

∫ (3x - 4).dx , x ∈ [1,3]

Answer 1

Let f(x) = 3x – 4, for 1 ≤ x ≤ 3

Divide the closed interval [1, 3] into n subintervals 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

∴ f(a + rh) = f(1 + rh)

= 3(1 + rh) – 4 

= 3rh – 1