Science:Math Exam Resources/Courses/MATH101/April 2005/Question 02 (c)

MATH101 April 2005

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Question 02 (c)
Let $\displaystyle R$ be the finite region bounded above by the curve $\displaystyle y=4-x^{2}$ and bounded below by $\displaystyle y=2-x$. Express the volume of the solid obtained by rotating $\displaystyle R$ about the vertical line $\displaystyle x=2$ as a definite integral. You do not need to simplify or evaluate this integral.

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
Consider what methods you have learned to evaluate the volumes of areas rotated about an axis. Is the axis you need to rotate about a horizontal or a vertical one?

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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. To evaluate this volume, we would use the method of cylindrical shells. We must determine first where the endpoints for the bounded area are. So we solve the following equation: ${\begin{aligned}4-x^{2}&=2-x\\0&=x^{2}-x-2\\0&=(x+1)(x-2)\quad \rightarrow \quad x=-1,2.\end{aligned}}$ For the method of cylindrical shells, we are basically summing up terms of the form $2\pi rh\Delta r$ where $r$ is the distance of each shell to the axis of rotation and $h$ is the height of the shell at radius $r$. The distance from the point $x$ to the axis of rotation is $r=|x-2|=2-x$ (since we are integrating over $[-1,2]$), and the height $h$ is just the difference of the functions that define the boundaries. Thus, the volume of the described object can be evaluating by the following integral: $V=\int _{-1}^{2}2\pi (2-x)(4-x^{2}-(2-x))\,dx$ (Note: Evaluating the volume gives $V=27\pi /2$.)

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