Science:Math Exam Resources/Courses/MATH101/April 2012/Question 08 (b)

MATH101 April 2012

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Question 08 (b)
Prove that for any positive real number a, we have $\int _{0}^{\pi /2}e^{-a\sin x}\,dx\leq {\frac {\pi }{2a}}$ Hint: First find a simple function ƒ(x) such that $\sin x\geq f(x)$ for $0\leq x\leq \pi /2$.

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
Following up on the hint already given, try a straight line as your simple function ƒ(x). It should be the steepest straight line that fits under $sin(x)$ between 0 and $\pi /2$.

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. The straight line below y=sin(x) that is selected for the inequality Following the hint, we want to find a straight line that is below $y=\sin(x)$ with minimal error. To do this, we require the straight line to pass through $y=\sin(x)$ at both $x=0$ and $x=\pi /2$, which are the endpoints of the domain of integration. Since $y=\sin(0)=0$ at $x=0$, we look for a straight line that passes through the origin with formula $y=f(x)=cx$. Then we put $x=\pi /2$, $y=\sin(\pi /2)=1$ to find $1=c\pi /2$, so $c=2/\pi$. Note that the slope of the line $c=2/\pi <1$, while the initial slope of $y=\sin(x)$ is given by $\cos(0)=1$, so the linear function $y=(2/\pi )x$ is really below $y=\sin(x)$ in the interval $[0,\pi /2].$ Hence, we can deduce the following inequalities: ${\begin{aligned}{\frac {2}{\pi }}x&\leq \sin x\\\Rightarrow -a\sin x&\leq -{\frac {2a}{\pi }}x\\\Rightarrow e^{-a\sin x}&\leq e^{-{\frac {2a}{\pi }}x}\\\end{aligned}}$ because $a>0$ and $e^{x}$ is a strictly increasing function. So by comparison, we obtain ${\begin{aligned}\int _{0}^{\pi /2}e^{-a\sin x}dx&\leq \int _{0}^{\pi /2}e^{-{\frac {2a}{\pi }}x}dx\\&=-{\frac {\pi }{2a}}\left[e^{-{\frac {2a}{\pi }}x}\right]_{0}^{\pi /2}\\&=-{\frac {\pi }{2a}}\left(e^{-a}-1\right)\\&={\frac {\pi }{2a}}\left(1-e^{-a}\right)<{\frac {\pi }{2a}}\end{aligned}}$ because $e^{-a}>0$ for any $a$.