Question

How do you use Riemann sums to evaluate the area under the curve of `f(x)= x^2` on the closed interval [1,3], with n=4 rectangles using midpoints?

Answer 1

`MRS = 8.75`

Explanation:

We have:

`f(x) = x^2`

We want to calculate over the interval `[1,3]` with `4` strips; thus:

`Deltax = (3-1)/4 = 0.5`

Note that we have a fixed interval (strictly speaking a Riemann sum can have a varying sized partition width). The values of the function are tabulated as follows;

Mid Riemann Sum

`MRS = sum_(r=0)^4 (f(x_i)+f(x_(i+1)))/2 \ Deltax_i`
`" " = 0.5 * (1.625 + 3.125 + 5.125 + 7.625)`
`" " = 0.5 * (17.5)`
`" " = 8.75`

Actual Value

For comparison of accuracy:

`Area = int_1^3 \ x^2 \ dx`
`" " = [x^3/3]_1^3`
`" " = 1/3[x^3]_1^3`
`" " = 1/3(27-1)`
`" " = 2/3`
`" " = 8.66666...`