Definite integral as a limit of sums
Definite integration · Mathematics
Study notes
Q: Express the integral from 0 to 2 of x^2 dx as a limit of sums and evaluate. Delta x = 2/n, x_i = 2i/n. Sum = (2/n) times sum of (2i/n)^2 = (2/n)(4/n^2) times n(n+1)(2n+1)/6. = (8/6n^3)(2n^3+3n^2+n) which tends to 16/6 = 8/3 as n grows large. So the integral equals 8/3. Matches x^3/3 from 0 to 2 = 8/3!