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

The body of a snowman is formed by the snowballs (this is its body) and (this is its head). a) Find the volume of the snowman by subtracting the intersection of the two snow balls from the sum of the volumes of the snow balls. Recall that the volume of a sphere of radius is . 
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 volume of the snowman is the volume of the body , the volume of the head minus the volume of the intesection . The hardest part is to find . To do so, we need to solve a triple integral. To find the boundaries for this triple integral, we make a sketch of the situation: 
Hint 2 

To find , we use the sketch from the previous hint and see that the region of integration is bounded below from and above from . 
Hint 3 

Finally, use cylindrical coordinates to integrate
Find the intersection of the two surfaces and solve for the radius, which will give you the upper limit for the radius. 
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.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. The volume of the snowman is the volume of the body , the volume of the head minus the volume of the intesection . First we calculate , which is the volume of the intersection of the two balls. The intersection is bounded below from the surface and above from . Since this shape is radially symmetric to the axes, we change to cylindrical coordinates. This means and .
Recall that the volume factor for cylindrical coordinates is .
The ball which makes the head has volume Hence, the volume of the snowman is
