Science:Math Exam Resources/Courses/MATH200/December 2011/Question 08 (b) ii

MATH200 December 2011

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Question 08 (b) ii

The body of a snowman is formed by the snowballs $x^{2}+y^{2}+z^{2}=12$ (this is its body) and $x^{2}+y^{2}+(z-4)^{2}=4$ (this is its head). b) We can also calculate the volume of the snowman as a sum of the following triple integrals: $(1)\quad \int _{0}^{\frac {2\pi }{3}}\int _{0}^{2\pi }\int _{0}^{2}\rho ^{2}\sin(\phi )\,d\rho \,d\theta \,d\phi$ $(2)\quad \int _{0}^{2\pi }\int _{0}^{\sqrt {3}}\int _{{\sqrt {3}}r}^{4-{\frac {r}{\sqrt {3}}}}r\,dz\,dr\,d\theta$ $(3)\quad \int _{\frac {\pi }{6}}^{\pi }\int _{0}^{2\pi }\int _{0}^{2{\sqrt {3}}}\rho ^{2}\sin(\phi )\,d\rho \,d\theta \,d\phi$ Circle the right answer from the underlined choices and fill in the blanks in the following descriptions of the region of integration for each integral. [Note: We have translated the axes in order to write down some of the integrals above. The equations you specify should be those before the translation is performed.] ii. The region of integration in (2) is a part of the snowman's body / head / body and head. It is enclosed by the sphere / cone defined by the equation                       and the sphere / cone defined by the equation

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
Consider which kind of variables (cartesian, cylindrical, spherical) are used in the integral.

Hint 2
The region of integration is enclosed in two cones. Use the upper and lower integration-limit of the variable $\displaystyle z$, as well as $\displaystyle r={\sqrt {x^{2}+y^{2}}}$ (cylindrical coordinates) to find the cone-equations.

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. Integral (2) is written in cylindrical coordinates, where the limit of the variables of integration is $\displaystyle \theta \in [0,2\pi ]$ for the horizontal angle, $\displaystyle r\in [0,{\sqrt {3}}]$ for the radius $\displaystyle r={\sqrt {x^{2}+y^{2}}}$ and $\displaystyle z\in \left[{\sqrt {3}}r,4-{\frac {r}{\sqrt {3}}}\right]$ for the vertical $\displaystyle z$ axis. Finding the equation of the enclosing cone from below The variables $\displaystyle r,\ \theta$ are in dependent and the angle covers the whole horizontal circle. The variable $\displaystyle z$ is depending or $\displaystyle r$. Hence, the region of integration is bounded from below by the equation $\displaystyle {\begin{aligned}z&={\sqrt {3}}r={\sqrt {3}}{\sqrt {x^{2}+y^{2}}}\\z^{2}&=3x^{2}+3y^{2}\end{aligned}}$ which is a cone-equation. Finding the equation of the enclosing cone from above Further, the region of integration is bounded from above by the equation $\displaystyle {\begin{aligned}z&=4-{\frac {r}{\sqrt {3}}}=4-{\frac {\sqrt {x^{2}+y^{2}}}{\sqrt {3}}}\\3(z-4)^{2}&=x^{2}+y^{2}\end{aligned}}$ which is also a cone-equation. Region of integration is part of body of head? The lower cone $\displaystyle z^{2}=3x^{2}+3y^{2}$ contains the point $\displaystyle (0,0,0)$ which is the center of the sphere $\displaystyle x^{2}+y^{2}+z^{2}=12$, the body of the snowman. Hence, a region above this cone is part of the body of the snowman. The upper cone $\displaystyle 3(z-4)^{2}=x^{2}+y^{2}$ contains the point $\displaystyle (0,0,4)$ which is the center of the sphere $\displaystyle x^{2}+y^{2}+(z-4)^{2}=4$, the head of the snowman. Hence, a region below this cone is part of the head of the snowman. So, the region of integration contains part of the head and the body of the snowman.

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