Science:Math Exam Resources/Courses/MATH101/April 2017/Question 07 (a)
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Question 07 (a) 

Let be the region to the right of the axis and to the left of the curve with between and . A reservoir dug into the ground has the shape that results from the region being rotated around the axis. (a) Calculate the volume of the reservoir (by slicing horizontally). Simplify your answer completely. 
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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! 
Hint 

Draw the region and the reservoir on  plane. Then, slice the reservoir horizontally. What's the volume of the slices? 
Checking a solution serves two purposes: helping you if, after having used the hint, 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 region and the reservoir are given on  plane as follows: Slicing the reservoir horizontally, for between and , we have a circle cross section at whose area is is . Combining with the height of each slice, we get the volume of the slice at as .
Here, the third equality follows from the substitution . Answer: 