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

MATH200 December 2011
Other MATH200 Exams

### Question 08 (b) iii

The body of a snowman is formed by the snowballs ${\displaystyle x^{2}+y^{2}+z^{2}=12}$ (this is its body) and ${\displaystyle x^{2}+y^{2}+(z-4)^{2}=4}$ (this is its head).

b) We can also calculate the volume of the snowman as a sum of the following triple integrals:

${\displaystyle (1)\quad \int _{0}^{\frac {2\pi }{3}}\int _{0}^{2\pi }\int _{0}^{2}\rho ^{2}\sin(\phi )\,d\rho \,d\theta \,d\phi }$
${\displaystyle (2)\quad \int _{0}^{2\pi }\int _{0}^{\sqrt {3}}\int _{{\sqrt {3}}r}^{4-{\frac {r}{\sqrt {3}}}}r\,dz\,dr\,d\theta }$
${\displaystyle (3)\quad \int _{\frac {\pi }{6}}^{\pi }\int _{0}^{2\pi }\int _{0}^{2{\sqrt {3}}}\rho ^{2}\sin(\phi )\,d\rho \,d\theta \,d\phi }$

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.]

iii. The region of integration in (3) 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

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