Antiderivative/integrals question

Question

Find a formula for the sum of n terms. Use the formula to find the limit as n → ∞.

Answer 1

Let f(x) = 7x², 0 ≤ x ≤ 1

Divide the interval [0,1] into n equal subintervals each of width Δx = 1/n.

Using left endpoints, the Riemann Sum approximation for ∫_{(0 to 1)}f(x)dx is :

∑_{(i = 1 to n)} [7((i-1)/n)²(1/n)] = ∑_{(i= 1 to n)} [7(i-1)² / n³]

lim_n→∞ ∑_{(I=1 to n)} [7(i-1)²/n³] = ∫_{(0 to 1)} 7x²dx = (7/3)x³_{(0 to 1)} = (7/3)(1 - 0) = 7/3

Answer 2

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