Answer
$10$
Work Step by Step
Let us consider $R_n$ be the approximation obtained by using $n$ rectangles of same width.
The exact area can be found under the graph of $f$:
$A=\lim\limits_{n\to \infty} f(x_k)\triangle x$; where $\triangle x=\dfrac{b-a}{n}$
Given: $f(x)=2x+1$; $x \in [1,3]$
Here, we have $\triangle x=\dfrac{b-a}{n}=\dfrac{3-1}{n}=\dfrac{2}{n}$
$A=\lim\limits_{n\to \infty}\sum_{k=1}^{n} f(x_k)\triangle x$
or, $A=\lim\limits_{n\to \infty}\sum_{k=1}^{n} [(\dfrac{6}{n}+\dfrac{8k}{n^2})]$
or, $A=\lim\limits_{n\to \infty}[6+4(1+\dfrac{1}{n})]$
Thus, $A=\lim\limits_{n\to \infty}[6+4+\dfrac{4}{n}]=10$
