MATH101 April 2010
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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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint

The right Riemann sum for the function $f(x)$ is:
$R_{n}=\sum _{i=1}^{n}f(x_{i})\Delta x$

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Solution

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The interval has size 1 so $\Delta x={\frac {1}{n}}$. The interval is $[0,1]$ so $x_{i}={\frac {i}{n}}$. In this case $f(x)=63x^{2}$ so the right Riemann sum is:
${\begin{aligned}R_{n}&=\sum _{i=1}^{n}(63({\frac {i}{n}})^{2})({\frac {1}{n}})\\&={\frac {6}{n}}\sum _{i=1}^{n}1{\frac {3}{n^{3}}}\sum _{i=1}^{n}i^{2}\\&={\frac {6}{n}}(n){\frac {3}{n^{3}}}{\frac {n(n+1)(2n+1)}{6}}\\&=6{\frac {n(n+1)(2n+1)}{2n^{3}}}\end{aligned}}$
So the integral is:
${\begin{aligned}\int _{0}^{1}(63x^{2})dx&=\lim _{n\to \infty }R_{n}\\&=\lim _{n\to \infty }(6{\frac {n(n+1)(2n+1)}{2n^{3}}})\\&=6\lim _{n\to \infty }{\frac {2n^{3}+3n^{2}+n}{2n^{3}}}\\&=6\lim _{n\to \infty }{\frac {2+3/n+1/n^{2}}{2}}\\&=6{\frac {200}{2}}=61=5\end{aligned}}$
Checking our answer, we see that
$\int _{0}^{1}(63x^{2})dx=(6xx^{3})\mid _{0}^{1}=6(1)(1)^{3}(6(0)(0)^{3})=61=5$
which matches the above.

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