Limit Formula to Evaluate Definite Integral as the Limit of a Sum

Limit Formula to Evaluate ^b∫_a f ( x)dx

Divide the interval [a , b] into n equal subintervals [ x₀, x₁ ], [ x₁, x₂ ], . . . , [ x_n-2, x_n-1 ], [ x_n-1, x_n ], such that a = x₀ < x₁ < x₂ < . . . < x_{n −1} < x_n = b . Then, we have x₁ – x₀ = x₂ – x₁ = . . . = x_n – x_n-1 = b-a / n . Put h = b−a / n. Then, we get x_i = a + ih, i = 1, 2, , n.

So, by the definition of definite integral, we get