Answer
$\dfrac{75}{2}$ or, $37.5$
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: $f(x)=3x$; $x \in [0,5]$
Here, we have $\triangle x=\dfrac{b-a}{n}=\dfrac{5-0}{n}=\dfrac{5}{n}$
$A=\lim\limits_{n\to \infty}\sum_{k=1}^{n} f(x_k)\triangle x$
or, $A=\lim\limits_{n\to \infty}\sum_{k=1}^{n} [(\dfrac{15k}{n})(\dfrac{5}{n})]$
or, $A=\lim\limits_{n\to \infty}[\dfrac{75}{n^2}\dfrac{n(n+1)}{2}]$
Thus, $A=\lim\limits_{n\to \infty}[\dfrac{75}{2}(1+\dfrac{1}{n})]=\dfrac{75}{2}=37.5$
