Science:Math Exam Resources/Courses/MATH200/April 2012/Question 10

MATH200 April 2012

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Question 10
Evaluate $\iiint \limits _{\mathbb {R} ^{3}}\left[1+(x^{2}+y^{2}+z^{2})^{3}\right]^{-1}{\textrm {d}}V.$

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
The combination $x^{2}+y^{2}+z^{2}$ suggests a change to spherical coordinates.

Hint 2
We integrate over the entire space, so what are the integration limits for ρ, θ and φ?

Hint 3
Don't forget to account for the differential volume element $\displaystyle {}dV$.

Hint 4
Luckily, the integrals with respect to φ and θ are easy to solve directly. For the final integration with respect to ρ use the substitution u = ρ3.

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. The combination $x^{2}+y^{2}+z^{2}$ suggests a change to spherical coordinates. The domain is compatible with this: in spherical coordinates, the whole space is covered when: θ runs from 0 to 2π, φ runs from 0 to π, and ρ runs from 0 to +∞. In spherical coordinates, the differential volume element is: $dV=\rho ^{2}\sin(\phi )\,d\rho \,d\theta \,d\phi$. And the function to integrate becomes: $(1+(x^{2}+y^{2}+z^{2})^{3})^{-1}=(1+(\rho ^{2})^{3})^{-1}={\frac {1}{1+\rho ^{6}}}$. Thus the integral of interest becomes: $I=\int _{\rho =0}^{\infty }\int _{\phi =0}^{\pi }\int _{\theta =0}^{2\pi }{\frac {\rho ^{2}\sin(\phi )}{1+\rho ^{6}}}d\theta \,d\phi \,d\rho =\int _{\rho =0}^{\infty }{\frac {\rho ^{2}}{1+\rho ^{6}}}\,d\rho \underbrace {\int _{\phi =0}^{\pi }\sin(\phi )\,d\phi } _{=2}\underbrace {\int _{\theta =0}^{2\pi }d\theta } _{=2\pi }$ Hence $I=4\pi \int _{\rho =0}^{\infty }{\frac {\rho ^{2}}{1+\rho ^{6}}}\,d\rho$ Substitute: $\displaystyle u=\rho ^{3}$ $\displaystyle du=3\rho ^{2}\,d\rho$ And you obtain: $I={\frac {4\pi }{3}}\int _{u=0}^{\infty }{\frac {du}{1+u^{2}}}={\frac {4\pi }{3}}{\Bigl .}\tan ^{-1}(u){\Bigr \vert }_{u=0}^{\infty }={\frac {2\pi ^{2}}{3}}.$