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

We split [2, 4] into $n$ subintervals of length $\dfrac{4-2}n=\dfrac2n$,

$[2,4]=\left[2,2+\dfrac2n\right]\cup\left[2+\dfrac2n,2+\dfrac4n\right]\cup\left[2+\dfrac4n,2+\dfrac6n\right]\cup\cdots\cup\left[2+\dfrac{2(n-1)}n,4\right]$

so that the right endpoints are given by the sequence

$x_i=2+\dfrac{2i}n=\dfrac{2(n+i)}n$

for $1\le i\le n$. Then the Riemann sum approximating

$\displaystyle\int_2^42x\,\mathrm dx$

is

$\displaystyle\sum_{i=1}^nf(x_i)\dfrac{4-2}n=\frac8{n^2}\sum_{i=1}^n(n+i)=\frac8{n^2}\left(n^2+\frac{n(n+1)}2\right)=\frac{12n+4}n$

The integral is given exactly as $n\to\infty$, for which we get

$\displaystyle\int_2^42x\,\mathrm dx=\lim_{n\to\infty}\frac{12n+4}n=12$

To check: we have

$\displaystyle\int_2^42x\,\mathrm dx=x^2\bigg|_2^4=4^2-2^2=16-4=12$