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)equals2x over the interval [2,4].

Answer 1

Given

$f(x)=2x,[2,4]$

Find

Area under the curve.

Explanation

first , we split interval [2 , 4] into n subintervals of length

$\frac{4-2}{n}=\frac{2}{n}$

so ,

$[2,4]=[2,2+\frac{2}{n}]\cup[2+\frac{2}{n},2+\frac{4}{n}]\cup[2+\frac{4}{n},2+\frac{6}{n}]\cup.....\cup[2+\frac{2(n-1)}{n},4]$

so that the right endpoints are given by the sequence

$x_i=2+\frac{2i}{n}=\frac{2(n+i)}{n},for\text{ }1\leq i\leq n$

then Riemann sum approximating ,

$\int_2^42xdx=\sum_{i\mathop{=}1}^nf(x_i)\frac{4-2}{n}=\frac{12n+4}{n}$

the integral is given exactly as n tends to infinity , for which we get ,

$\int_2^42xdx=\lim_{n\to\infty}(\frac{12n+4}{n})=12$

Final Answer

Hence ,

$\int_2^42xdx=12$