Question

How do you use a Riemann sum to calculate a definite integral?

Answer 1

Definition of Definite Integral

`int_a^b f(x) dx=lim_{n to infty}sum_{i=1}^n f(x_i) Delta x`,

where `x_i=a+iDelta x` and `Delta x={b-a}/n`.

Let us look at the following example.

`int_1^3(2x+1)dx`

by definition,

`=lim_{n to infty}sum_{i=1}^n[2(1+2/ni)+1]2/n`

by simplifying the expression inside the summation,

`=lim_{n to infty}sum_{i=1}^n(8/n^2i+6/n)`

by splitting the summation and pulling out constants,

`=lim_{n to infty}(8/n^2sum_{i=1}^ni+6/nsum_{i=1}^n1)`

by the summation formulas `sum_{i=1}^ni={n(n+1)}/2` and `sum_{i=1}^n1=n`,

`=lim_{n to infty}(8/n^2cdot{n(n+1)}/2+6/ncdot n)`

by cancelling out `n`'s,

`=lim_{n to infty}[4(1+1/n)+6]=4(1+0)+6=10`

I hope that this was helpful.