Use A=lim(n→∞)Rn=lim(n→∞)[f(x1)Δx+...f(xn)Δx] to find an expression for the area under the graph of f as a limit. Do not evaluate the limit.

Question

(2x)/(x²+1), 1≤x≤3

The area of the region that lies under the graph of the continuous function, f, is the limit of the sum of the areas of approximating rectangles. A=lim_(n→∞)R_n=lim_(n→∞)[f(x₁)Δx+...f(x_n)Δx]

Answer 1

This Riemann sum is equal to the area under the curve of f(x)=2x/x^2+1 from 1 to 3, which in turn is equal to ∫f(x)dx from 1 to 3. By puting u=x^2+1 this integral changes to ∫du/u from 2 to 10 which is equal to: ln5.