Science:Math Exam Resources/Courses/MATH101/April 2011/Question 06 (b)

MATH101 April 2011

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Question 06 (b)
Use part (a) to determine an approximation of the following definite integral $\int _{0}^{0.1}{\frac {1}{1+x^{3}}}dx$ so that the error in the approximation satisfies $|{\text{error}}|\leq 10^{-9}$ Remember to justify your answer.

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
With a convergent power series, you can integrate term by term.

Hint 2
For an alternating series, the error is less than or equal to the absolute value of the first neglected term.

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

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 saw from part (a) that we have the series expansion ${\frac {1}{1+x^{3}}}=1-x^{3}+x^{6}-x^{9}+\ldots$ and so we can write that ${\begin{aligned}\int _{0}^{0.1}{\frac {1}{1+x^{3}}}\,dx&=\int _{0}^{0.1}dx-\int _{0}^{0.1}x^{3}\,dx+\int _{0}^{0.1}x^{6}\,dx-\int _{0}^{0.1}x^{9}\,dx+\ldots \\&=0.1-{\frac {(0.1)^{4}}{4}}+{\frac {(0.1)^{7}}{7}}-{\frac {(0.1)^{10}}{10}}+\ldots \end{aligned}}$ where the $(n+1)$st term is ${\frac {(-1)^{n}(0.1)^{3n+1}}{3n+1}}$ Since this is an alternating sequence which converges to zero, we have that the error when neglecting everything past the $(n+1)$st term. Thus ${\begin{aligned}|{\textrm {error}}|&\leq \left|{\frac {(-1)^{n}(0.1)^{3n+1}}{3n+1}}\right|\\&={\frac {0.1^{3n+1}}{3n+1}}\\&={\frac {10^{-(3n+1)}}{3n+1}}\end{aligned}}$ if we want this to be less than $10^{-9}$, then we can choose $n=3$, where the error will be less than ${\frac {10^{-(9+1)}}{9+1}}=10^{-11}$ which works fine since it is clearly even less than $10^{-9}$.

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