Find area of a limit off region

Question

Use the limit process to find The area of the region between f(x)=x^2-x and the x-axis on the interval [1,3]

Answer 1

f(x)=x²-x. The area is calculated as ∫₁³f(x)dx. By the definition of the definite integral ∫13f(x)dx=lim_n->∞∑_i=1ⁿf(x_i)Δx_i.

First, we can’t actually use this definition unless we determine which points in each interval that well use for. To make things easier we’ll use the right endpoints of each interval. 

Δx_i=Δx=(3-1)/n=2/n. The subintervals are

[1, 1+2/n], [1+2/n, 1+4/n],...,[1+2(i-1)/n, 1+2i/n],....,[1+2(n-1)/n, 3]

we take x_i^*=1+2i/n.

∑_i=1ⁿf(x_i^*)Δxi=∑_i=1ⁿf(1+2i/n)(2/n)=∑_i=1ⁿ[(1+2i/n)²-(1+2i)/n](2/n)=

=(4/n²)∑_i=1ⁿi(1+2i/n)=2(n+1)/n+(4/3)(n+1)(2n+1)=S_n

lim_n->∞S_n=14/3. This is the requared area. 

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