Integral Calculus

Question

1.The area A of a region that lies under the graph of the continuous function f is given by A = limn→∞ Rn = limn→∞ [f(x1)∆x + f(x2)∆x + · · · + f(xn)∆x].

(a) Use this definition to find an expression for the area under the curve y = x^3 from 0 to 1 as a limit.

1. Use the formula 1 3 + 23 + 33 + · · · + n 3 = n(n + 1) 2 2 to evaluate the limit in part (a)

Answer 1

R_n = ∑_j=1ⁿf(x_j)Δx

= ∑_j=1ⁿf((j–1)Δx)Δx

= ∑_j=1ⁿf((j–1)/(n–1))/(n–1)

= ∑_j=2ⁿ(j–1)³/(n–1)⁴

= ∑_k=1^n–1k³/(n–1)⁴

= (n–1)²n²/4(n–1)⁴

= (n/(n–1))²/4

→_n→∞ 1/4.