MATH105 April 2013
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) • Q1 (f) • Q1 (g) • Q1 (h) • Q1 (i) • Q1 (j) • Q1 (k) • Q1 (l) • Q1 (m) • Q1 (n) • Q2 (a) • Q2 (b) • Q2 (c) • Q3 (a) • Q3 (b) • Q4 (a) • Q4 (b) • Q5 (a) • Q5 (b) • Q6 (a) • Q6 (b) •
Question 01 (c)

ShortAnswer Questions. Put your answer in the box provided but show your
work also. Each question is worth 3 marks, but not all questions are of equal difficulty.
Evaluate $\displaystyle \int _{0}^{5/2}{\frac {dx}{\sqrt {25x^{2}}}}$.

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

The identity $\displaystyle {}\sin ^{2}\theta +\cos ^{2}\theta =1$ may be useful here.

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.
We start by noting that the denominator is in the form
 $\displaystyle {}a^{2}x^{2}$
with a = 5. This is a good indicator that we want to do a trig substitution with
 $\displaystyle {}x=a\sin \theta$
because
 $\displaystyle {}a^{2}\sin ^{2}\theta +a^{2}\cos ^{2}\theta =a^{2}$
and hence
 $\displaystyle {}a^{2}a^{2}\sin ^{2}\theta =a^{2}\cos ^{2}\theta$
We will use this identity to handle the square root in the denominator. So, let
 ${\begin{aligned}x&=5\sin \theta \\dx&=5\cos \theta d\theta .\end{aligned}}$
We must also modify the limits of integration since we have a new variable. When
 ${\begin{aligned}x&=0=5\sin \theta \end{aligned}}$
solving for $\theta$ gives $\displaystyle {}\theta =0$. Solving
 ${\begin{aligned}x&=5/2=5\sin \theta \end{aligned}}$
for $\theta$ gives $\displaystyle {}\theta ={\frac {\pi }{6}}$. Substituting these values into our integral we obtain
 $\int _{0}^{5/2}{\frac {dx}{\sqrt {25x^{2}}}}=\int _{0}^{\pi /6}{\frac {5\cos \theta d\theta }{\sqrt {25(5\sin \theta )^{2}}}}=\int _{0}^{\pi /6}{\frac {5\cos \theta d\theta }{\sqrt {(5)^{2}(5)^{2}(\sin \theta )^{2}}}}.$
We can use the identity above we get
 $\int _{0}^{\pi /6}{\frac {5\cos \theta d\theta }{\sqrt {(5)^{2}(5)^{2}(\sin \theta )^{2}}}}=\int _{0}^{\pi /6}{\frac {5\cos \theta d\theta }{\sqrt {(5)^{2}(\cos \theta )^{2}}}}=\int _{0}^{\pi /6}1d\theta ={\frac {\pi }{6}}$
Since this is a definite integral, the result is a number and we do not need to return to the previous variables.

Click here for similar questions
MER QGH flag, MER QGQ flag, MER QGS flag, MER RT flag, MER Tag Trigonometric substitution, 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
