Question

How do you calculate the left and right Riemann sum for the given function over the interval [0,2], n=4 for `f(x) = (e^x) − 5`?

Answer 1

`L RS = -5.0757`
`R RS = -1.8811`

Explanation:

We have:

`f(x) = e^x-5`

graph{e^x-5 [-10, 10, -8, 8]}

We want to calculate over the interval `[0,2]` with `4` strips; thus:

`Deltax = (2-0)/4 = 1/2`

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`
`" " = 0.5 * (-4 - 3.3513 - 2.2817 - 0.5183)`
`" " = 0.5 * (-10.1513)`
`" " = -5.0757`

Right Riemann Sum

`R RS = sum_(r=1)^4 f(x_i)Deltax_i`
`" " = 0.5 * (-3.3513 - 2.2817 - 0.5183 + 2.3891)`
`" " = 0.5 * (-3.7623)`
`" " = -1.8811`

Actual Value

For comparison of accuracy:

`Area = int_0^2 \ e^x-5 \ dx`
`" " = [e^x-5x]_0^2`
`" " = (e^2-10) - (e^0-0)`
`" " = e^2-10 - 1`
`" " = e^2-11`
`" " = -3.610943 ...`