Question

How do you use Riemann sums to evaluate the area under the curve of `f(x)= 3x` on the closed interval [1,5], with n=4 rectangles using right, left, and midpoints?

Answer 1

`L RS = 30`
`R RS = 42`
`M RS = 36`

Explanation:

We have:

`f(x) = 3x`

We want to calculate over the interval `[1,5]` with `4` strips; thus:

`Deltax = (5-1)/4 = 1`

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;

Left Riemann Sum

`L RS = sum_(r=0)^3 f(x_i)Deltax_i`
`" " = 1 * (3 + 6 + 9 + 12)`
`" " = 1 * (30)`
`" " = 30`

Right Riemann Sum

`R RS = sum_(r=1)^4 f(x_i)Deltax_i`
`" " = 1 * (6 + 9 + 12 + 15)`
`" " = 1 * (42)`
`" " = 42`

Mid Riemann Sum

`M RS = sum_(r=0)^3 ((f(x_i)+f(x_(i+1)))/2)Deltax_i`
`" " = 1 * (4.5 + 7.5 + 10.5 + 13.5)`
`" " = 1 * (36)`
`" " = 36`

Actual Value

For comparison of accuracy:

`Area = int_1^5 \ 3x \ dx`
`" " = [(3x^2)/2]_1^5`
`" " = 3/2 \ [x^2]_1^5`
`" " = 3/2( 25-1)`
`" " = 36`