Question

How do you use Riemann sums to evaluate the area under the curve of `f(x) = (e^x) − 5` on the closed interval [0,2], with n=4 rectangles using midpoints?

Answer 1

the answer
`S_p=-3.67701446661601`

Explanation:

The sketch of our function `f(x) = (e^x) − 5`

graph{e^x-5 [-16.02, 16.02, -8.01, 8.01]}

the width

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

The midpoints

`(0+1/2)/2=1/4`
`(1/2+1)/2=3/4`
`(1+3/2)/2=5/4`
`(3/2+2)/2=7/4`

now find the high
`f(1/4)=-3.71597458331226`
`f(3/4)=-2.88299998338733`
`f(5/4)=-1.50965704253816`
`f(7/4)=0.75460267600573`

The sketch of our function with midpoints

calculate Riemann sum

`S_p=width*high`

`S_p=(1/2)[-3.71597458331226-2.88299998338733-1.50965704253816+0.75460267600573]`

`S_p=-3.67701446661601`