*Use the form of the definition of the integral given in the theorem to evaluate the integral.

Question

Use the form of the definition of the integral given in the theorem to evaluate the integral.

Integral -2 to 0 of (4x^2 + 4x) dx

0
(4x2 + 4x) dx
−2

 

Answer 1

Δx = (0-(-2))/n = 2/n

Use ∑1 = n, ∑i = n(n+1)/2 and ∑i² = n(n+1)(2n+1)/n

The summation equivalent of the integral is:

4[∑(-2+2i/n)² -2 +2i/n](2/n) = 4[∑4-8i/n+4i²/n² -2 +2i/n](2/n) = 4[∑2-6i/n+4i²/n²](2/n)

=4[4-6n(n+1)/n² + (8/6)n(n+1)(2n+1)/n³]

Taking the limit as n→∞

4[4-6+(8/6)2] = 4[-2+8/3] = 8/3