Science:Math Exam Resources/Courses/MATH101 C/April 2024/Question 03

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MATH101 C April 2024

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Other MATH101 C Exams

• April 2024 •

Question 03
Evaluate $\int {\sqrt {9-x^{2}}}dx.$

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
Because the integrand involves a difference of squares, try using a trigonometric substitution.

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
If we imagine that the integrand is the side of a right-angled triangle, then the other sides of the triangle have length $x$ and 3, and 3 is the length of the hypotenuse. We use this picture to guess the following substitution: ${\begin{aligned}{\frac {x}{3}}&=\sin(\theta )\\\mathrm {d} x&=3\cos(\theta )\mathrm {d} \theta \end{aligned}}$ We thus have $\int {\sqrt {9-x^{2}}}\mathrm {d} x=\int {\sqrt {9-9\sin ^{2}(\theta )}}\cdot 3\cos(\theta )\mathrm {d} \theta =\int 9\cos ^{2}(\theta ).\mathrm {d} \theta$ We now use the trigonometric identity $\cos ^{2}(\theta )={\frac {1+\cos(2\theta )}{2}}$ to evaluate $\int 9\cos ^{2}(\theta )\mathrm {d} \theta ={\frac {9}{2}}\int \left(1+\cos(2\theta )\right)\mathrm {d} \theta ={\frac {9}{2}}\left(\theta +{\frac {\sin(2\theta )}{2}}\right)+C.$ We have $\theta =\arcsin \left({\frac {x}{3}}\right)$ , so, we can rewrite the answer in terms of $x$ : $\int {\sqrt {9-x^{2}}}\mathrm {d} x={\frac {9}{2}}\left(\arcsin \left({\frac {x}{3}}\right)+{\frac {\sin \left(2\arcsin \left({\frac {x}{3}}\right)\right)}{2}}\right)+C={\frac {9}{2}}\arcsin \left({\frac {x}{3}}\right)+{\frac {9}{4}}\left(2\sin \left(\arcsin \left({\frac {x}{3}}\right)\right)\cos \left(\arcsin \left({\frac {x}{3}}\right)\right)\right)+C,$ where, in the last step, we used the trigonometric formula $\sin(\alpha +\beta )=\sin(\alpha )\cos(\beta )+\sin(\beta )\cos(\alpha )$ with $\alpha =\beta =\arcsin(x/3)$ . By definition of the arcsine, $\sin \left(\arcsin \left({\frac {x}{3}}\right)\right)={\frac {x}{3}}$ . But this is also to be expected, given our interpretation of $x$ as the length of the side opposite to $\theta$ in a right-angled triangle with hypotenuse of length 3. This picture also shows that $\cos \left(\arcsin \left({\frac {x}{3}}\right)\right)={\frac {\sqrt {9-x^{2}}}{3}}.$ Putting it all together, we find $\int {\sqrt {9-x^{2}}}\mathrm {d} x={\frac {9}{2}}\arcsin \left({\frac {x}{3}}\right)+{\frac {9}{2}}\left({\frac {x}{3}}\cdot {\frac {\sqrt {9-x^{2}}}{3}}\right)+C,$ which is an equation that makes sense for $-3\leq x\leq 3$ .