Answer
$\dfrac{223}{35}$ or, $6.371$
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: $f(x)=\dfrac{4}{x}$
Here, we have $\triangle x=\dfrac{b-a}{n}=\dfrac{7-1}{6}=1$
$A=\lim\limits_{n\to \infty} f(x_k)\triangle x$
or, $A=[f(x_1)+f(x_2)+f(x_3)+f(x_4)]\triangle x$
or, $A=[\dfrac{4}{2}+\dfrac{4}{3}+\dfrac{4}{4}+\dfrac{4}{5}+\dfrac{4}{6}+\dfrac{4}{7}](0.5)$
Thus, $A=\dfrac{223}{35}$ or, $6.371$
