Science:Math Exam Resources/Courses/MATH105/April 2013/Question 03 (b)

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MATH105 April 2013

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Question 03 (b)
Let $f(0)=1,\,f(2)=3$ and $\displaystyle f'(2)=4$. Calculate $\int _{0}^{4}f''({\sqrt {x}})\,dx$ .

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
Start with the substitution $u={\sqrt {x}}$

Hint 2
There are two techniques to reduce the order of the derivative under the integral: integration by parts and the fundamental theorem of calculus. You want to use both, in the right order.

Checking a solution serves two purposes: helping you if, after having used all the hints, 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

We begin evaluating this integral by making a change of variable to deal with the square root: $u={\sqrt {x}}\quad \rightarrow \quad du={\frac {1}{2{\sqrt {x}}}}\,dx={\frac {1}{2u}}\,dx.$ And hence, $\int _{0}^{4}f''({\sqrt {x}})\,dx=2\int _{0}^{2}uf''(u)\,du.$ Now using integration by parts we evaluate the integral. Let $\displaystyle v=u$ and $\displaystyle dw=f''(u)\,du$ so that we have $\displaystyle dv=du$ and $\displaystyle w=f'(u)$ (which holds by the Fundamental Theorem of Calculus as $\displaystyle f'(u)$ is an antiderivative of $\displaystyle f''(u)$. Thus ${\begin{aligned}2\int _{0}^{2}uf''(u)\,du&=2\left(uf'(u){\Big \vert }_{0}^{2}-\int _{0}^{2}f'(u)\,du\right)\end{aligned}}$ Now, we once again use the Fundamental Theorem of Calculus and note that $\displaystyle f(u)$ is an antiderivative of $\displaystyle f'(u)$ and this gives ${\begin{aligned}2\left(uf'(u){\Big \vert }_{0}^{2}-\int _{0}^{2}f'(u)\,du\right)&=2\left(uf'(u)-f(u)\right){\Big \vert }_{0}^{2}\\&=2\left(2f'(2)-f(2)+f(0)\right)\end{aligned}}$ Plugging the values of the terms given in the statement of the question gives: ${\color {blue}\int _{0}^{4}f''({\sqrt {x}})\,dx=2\left(2(4)-3+1\right)=12.}$

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