MATH101 April 2018
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Question 04 (i)

Write an integral that corresponds to the limit as ${\textstyle n\to \infty }$ of the Riemann sum $\sum _{i=1}^{n}{\frac {4i}{n^{2}}}e^{2i/n}\,.$ Then calculate the resulting integral explicitly.

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1

Start with $x_{i}={\frac {i}{n}}$.

Hint 2

Consider the integration by parts to evaluate the integral.

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Solution

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As suggested in the Hint, start with $x_{i}={\frac {i}{n}}$ and $\Delta x=x_{i+1}x_{i}={\frac {1}{n}}$. Then, the Riemann sum can be written as
$\sum _{i=1}^{n}{\frac {4i}{n^{2}}}e^{2i/n}=\sum _{i=1}^{n}4x_{i}e^{2x_{i}}\Delta x.$
Observe that $x_{1}={\frac {1}{n}}\to 0$ as $n\to \infty$ and $x_{n}=1$ for any $n\in \mathbb {N}$. Based on this, we put $f(x)=4xe^{2x}$ to get
$\sum _{i=1}^{n}4x_{i}e^{2x_{i}}\Delta x=\sum _{i=1}^{n}f(x_{i})\Delta x\to \int _{0}^{1}f(x)dx=\int _{0}^{1}4xe^{2x}dx,$
as $n\to \infty$.
Now, we evaluate the integral. Applying the integration by parts with $f(x)=4x$, $g(x)=e^{2x}$, $f'(x)=4$, and $g'(x)=2e^{2x}$, we have
${\begin{aligned}\int _{0}^{1}4xe^{2x}dx&={\frac {1}{2}}\int _{0}^{1}f(x)g'(x)dx={\frac {1}{2}}\left[{\bigg (}f(x)g(x){\bigg }_{0}^{1}\int _{0}^{1}f'(x)g(x)dx\right]\\&={\frac {1}{2}}(4e^{2}0)+{\frac {1}{2}}\int _{0}^{1}4e^{2x}dx=2e^{2}+2\int _{0}^{1}e^{2x}dx\\&=2e^{2}+2{\bigg (}{\frac {1}{2}}e^{2x}{\bigg }_{0}^{1}\\&=2e^{2}+2\cdot ({\frac {1}{2}})\cdot {\bigg (}e^{2x}{\bigg }_{0}^{1}\\&=2e^{2}(e^{2}e^{0})=3e^{2}+1.\end{aligned}}$
Answer: $\color {blue}3e^{2}+1$

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