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]

$h=\frac{b-a}{n}=\frac{5-2}{n}=\frac{3}{n}$

$x_i=i\frac{3}{n}$

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

$f(x_i)=4i\times \frac{3}{n}=\frac{12i}{n}$

Riemann sum=$\lim_{n\rightarrow \infty}\sum_{i=1}^{n}f(x_i)\cdot h=\lim_{n\rightarrow \infty}\sum_{i=1}^{n}(\frac{12i}{n}\times \frac{3}{n}$

Riemann sum=$\lim_{n\rightarrow \infty}\frac{36}{n^2}\sum_{i=1}^{n}i$

Riemann sum=$\lim_{n\rightarrow \infty}\frac{36}{n^2}\times \frac{n(n+1)}{2}$

By using

$\sum n=\frac{n(n+1)}{2}$

Riemann sum=$\lim_{n\rightarrow \infty}\frac{18n(n+1)}{n^2}=\lim_{n\rightarrow \infty}18(1+\frac{1}{n})$

Apply the limit

Area under the curve=$18$ square units