Science:Math Exam Resources/Courses/MATH200/December 2011/Question 07

MATH200 December 2011

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Question 07
Evaluate the triple integral $\iiint _{E}x\,dV$, where E is the region in the first octant bounded by the parabolic cylinder $\displaystyle y=x^{2}$ and the planes $y+z=1,\,x=0,\,z=0$.

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
The hardest part here is to find the boundaries for the region of integration. Often it is very helpful to make a sketch, like this (boundaries of the region are in red):

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. 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. The plane $\displaystyle z+y=1$ intesects $\displaystyle z=0$ at the line $\displaystyle y=1,z=0$. Since the region is is the first octant, we know that $\displaystyle x,y,z\geq 0$. Using this information and the sketch in the hint, we find the region $\displaystyle E$ as $\displaystyle E=\{(x,y,z)\in \mathbb {R} ^{3}|\ 0\leq z\leq 1-y,\ x^{2}\leq y\leq 1,\ 0\leq x\leq 1\}$ Hence the integral is $\displaystyle {\begin{aligned}\iiint _{E}x\;{\text{d}}V&=\int _{0}^{1}\int _{x^{2}}^{1}\int _{0}^{1-y}x\;{\text{d}}z{\text{d}}y{\text{d}}x\\&=\int _{0}^{1}\int _{x^{2}}^{1}x(1-y)\;{\text{d}}y{\text{d}}x\\&=\int _{0}^{1}x\left[y-{\frac {y^{2}}{2}}\right]_{x^{2}}^{1}\;{\text{d}}x\\&=\int _{0}^{1}{\frac {x}{2}}-x^{3}+{\frac {x^{5}}{2}}\;{\text{d}}x\\=&\left[{\frac {x^{2}}{4}}-{\frac {x^{4}}{4}}+{\frac {x^{6}}{12}}\right]_{0}^{1}={\frac {1}{12}}\end{aligned}}$

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