Science:Math Exam Resources/Courses/MATH101/April 2009/Question 01 (b)/Solution 1

We recognize that this has the form of a Riemann sum:

$\lim _{n\rightarrow \infty }\sum _{i=1}^{n}f(x_{i})\Delta x$

We identify

$\Delta x={\frac {b-a}{n}}={\frac {\pi }{4n}}$

and

$x_{i}=a+i\Delta x=a+{\frac {i\pi }{4n}}.$

This gives us that

$b-a={\frac {\pi }{4}}$

Now, notice that the function in question is $f(x)=\tan(x)$ and so

${\begin{aligned}f(x_{i})&=\tan(x_{i})\\&=\tan \left(a+{\frac {i\pi }{4n}}\right)\\&=\tan \left({\frac {i\pi }{4n}}\right)\end{aligned}}$

and so we see that we can choose $a=0$. This gives us that

$b=b-0=b-a={\frac {\pi }{4}}.$

Hence, the integral is

$\int _{0}^{\tfrac {\pi }{4}}\tan(x)\,dx$

completing the question.