Question

For the function given below, find a formula for the Riemann sum obtained by dividing the interval [a,b] into n equal subintervals and using the right-hand endpoint for each ck. Then take a limit of this sum as n-oo to calculate the area under the curve over [a,b] 10x+103 over the intervall -10 Find a formula for the Riemann sum.

Part 2: Calculate the area under the curve.

Answer 1

Given that delta_x = 0 + 1/n = 1/n, f(x_i) = 10 (-1 + i/n)^2 + 10 (-1 + i/n)^3 = 10 (1 - 2i/n + i^2/n^2) + 10 (-1 + 3i/n - 3i^2/n^2 + i^3/n^3) = 10i/n - 20i^2/n^2 + 10i^3/n^3. S_n = sigma [10 i/n - 20i^2/n^2 + 10i^3/n^3]^1/n. Area = lim_n rightarrow infinity sigma_l = 1 [10 i/n - 20i^2/n^2 + 10i^3/n^3]^1/n = lim_n rightarrow infinity [10 n(n + 1)/2n^2 -20 n(n + 1) (2n + 1)/6n^3 + 10 n^2 (n + 1)^2/4n^4] = lim_n rightarrow infinity [5 (1 + 1/n) - 20 (1 + 1/n) (2 + 1/n) times 1/6 + 10 times 1/4 (1 + 1/n)^2] = 5 - 20/3 + 5/2 = 30 - 40 + 15/6 = 5/6 Area = 5/6.