Question

For the function given​ below, find a formula for the Riemann sum obtained by dividing the interval​ [a,b] into n equal subintervals and using the​ right-hand endpoint for each c Subscript k. Then take a limit of this sum as n right arrow infinity to calculate the area under the curve over​ [a,b]. ​f(x)equals4x over the interval ​[2​,5​]. Find a formula for the Riemann sum.

Answer 1

Answer with Step-by-step explanation:

We are given that

`f(x)=4x`

Interval=[2,5]

`h=(b-a)/(n)=(5-2)/(n)=(3)/(n)`

`x_i=i(3)/(n)`

Where i=1,2,3,... n

`f(x_i)=4i* (3)/(n)=(12i)/(n)`

Riemann sum=
`\lim_(n\rightarrow \infty)\sum_(i=1)^(n)f(x_i)\cdot h=\lim_(n\rightarrow \infty)\sum_(i=1)^(n)((12i)/(n)* (3)/(n)`

Riemann sum=
`\lim_(n\rightarrow \infty)(36)/(n^2)\sum_(i=1)^(n)i`

Riemann sum=
`\lim_(n\rightarrow \infty)(36)/(n^2)* (n(n+1))/(2)`

By using

`\sum n=(n(n+1))/(2)`

Riemann sum=
`\lim_(n\rightarrow \infty)(18n(n+1))/(n^2)=\lim_(n\rightarrow \infty)18(1+(1)/(n))`

Apply the limit

Area under the curve=
`18` square units