MATH101 April 2014
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) • Q1 (f) • Q1 (g) • Q1 (h) • Q1 (i) • Q1 (j) • Q1 (k) • Q2 • Q3 • Q4 • Q5 • Q6 • Q7 (a) • Q7 (b) • Q8 • Q9 (a) • Q9 (b) •
Question 05

Long Problem. Show your work. No credit will be given for the answer without the correct accompanying work.
Find the centroid of the finite region bounded by $\displaystyle y=\sin(x),\ y=\cos(x),\ x=0$ and $\displaystyle x=\pi /4$.

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

Use the formula for the centroid:
${\begin{aligned}{\bar {x}}={\frac {1}{A}}\int _{a}^{b}x(f(x)g(x))dx,\quad {\text{and}}\quad {\bar {y}}={\frac {1}{2A}}\int _{a}^{b}[f(x)]^{2}[g(x)]^{2}dx\end{aligned}}$
where $A$ is the area of the region.

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
 If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work.
 If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies.
First, the given region is the following:
On the region of interest $0\leq x\leq {\frac {\pi }{4}}$, we have $\cos(x)\geq \sin(x)$. By the formula for the centroid, we have
 ${\begin{aligned}{\bar {x}}={\frac {1}{A}}\int _{0}^{\frac {\pi }{4}}x(\cos x\sin x)dx,\quad {\text{and}}\quad {\bar {y}}={\frac {1}{2A}}\int _{0}^{\frac {\pi }{4}}\cos ^{2}x\sin ^{2}xdx\end{aligned}}$
where $A$ is the area of the region which we will compute first:
 ${\begin{aligned}A=\int _{0}^{\frac {\pi }{4}}\cos x\sin xdx=\left[\sin x+\cos x\right]_{0}^{\frac {\pi }{4}}={\frac {1}{\sqrt {2}}}+{\frac {1}{\sqrt {2}}}1={\sqrt {2}}1.\end{aligned}}$
Next we will compute the integral for the $x$ portion of the centroid,
 $\int _{0}^{\frac {\pi }{4}}x(\cos x\sin x){\textrm {d}}x=\int _{0}^{\frac {\pi }{4}}x\cos x{\textrm {d}}x\int _{0}^{\frac {\pi }{4}}x\sin x{\textrm {d}}x.$
We integrate each term by parts. For the first integral,
 $\int _{0}^{\frac {\pi }{4}}x\cos x{\textrm {d}}x=\left[x\sin x\right]_{0}^{\frac {\pi }{4}}\int _{0}^{\frac {\pi }{4}}\sin x{\textrm {d}}x=\left[x\sin x\right]_{0}^{\frac {\pi }{4}}+\left[\cos x\right]_{0}^{\frac {\pi }{4}}={\frac {\pi }{4}}{\frac {1}{\sqrt {2}}}+{\frac {1}{\sqrt {2}}}1.$
For the second integral,
 $\int _{0}^{\frac {\pi }{4}}x\sin x{\textrm {d}}x=\left[x\cos x\right]_{0}^{\frac {\pi }{4}}+\int _{0}^{\frac {\pi }{4}}\cos x{\textrm {d}}x=\left[x\cos x\right]_{0}^{\frac {\pi }{4}}+\left[\sin x\right]_{0}^{\frac {\pi }{4}}={\frac {\pi }{4}}{\frac {1}{\sqrt {2}}}+{\frac {1}{\sqrt {2}}}.$
subtracting the two integrals we have
 $\int _{0}^{\frac {\pi }{4}}x(\cos x\sin x){\textrm {d}}x={\frac {\pi }{2{\sqrt {2}}}}1.$
Next we need to consider the integral for the $y$ component of the centroid,
 $\int _{0}^{\frac {\pi }{4}}\cos ^{2}x\sin ^{2}x{\textrm {d}}x=\int _{0}^{\frac {\pi }{4}}\cos 2x{\textrm {d}}x,$
where we have used the trigonometric identity $\cos ^{2}x\sin ^{2}x=\cos 2x$. Therefore we have
 $\int _{0}^{\frac {\pi }{4}}\cos 2x{\textrm {d}}x=\left[{\frac {1}{2}}0\right]={\frac {1}{2}}.$
Thus, the centroid $({\bar {x}},{\bar {y}})$ is
${\begin{aligned}{\bar {x}}={\frac {1}{A}}\int _{0}^{\frac {\pi }{4}}x(\cos x\sin x)dx,={\frac {1}{{\sqrt {2}}1}}\left({\frac {\pi }{2{\sqrt {2}}}}1\right)\end{aligned}}$
and
${\begin{aligned}{\bar {y}}={\frac {1}{2A}}\int _{0}^{\frac {\pi }{4}}\cos ^{2}x\sin ^{2}xdx={\frac {1}{2({\sqrt {2}}1)}}\cdot {\frac {1}{2}}={\frac {1}{4({\sqrt {2}}1)}}.\end{aligned}}$

Click here for similar questions
MER QGH flag, MER QGQ flag, MER QGS flag, MER QGT flag, MER Tag Area between two curves, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag

Math Learning Centre
 A space to study math together.
 Free math graduate and undergraduate TA support.
 Mon  Fri: 12 pm  5 pm in LSK 301&302 and 5 pm  7 pm online.
Private tutor
