Using the RN approximation, express the area under the graph f(x) = x^−3 over [1, 9] as a limit but do not evaluate.

Question

I don't really understand integrals and summation.

Answer 1

Δx = (9-1)/n = 8/n

Divide the interval [1,9] into n equal subintervals each of width Δx.

Right endpoint of first interval = 1 + Δx

Right endpoint of 2nd interval = 1 + 2Δx

Right endpoint of 3rd interval = 1 + 3Δx......

Right endpoint of last interval = 1 + nΔx = 9

R_n = Σ_{(i = 1 to n)} [f(1 + iΔx)Δx]

∫_{(from 1 to 9)}f(x)dx = lim_n→∞R_n =

lim_n→∞[∑_{(i = 1 to n)}{(1 + 8i/n)^-3(8/n)}]