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):

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Solution

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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
$\displaystyle E=\{(x,y,z)\in \mathbb {R} ^{3}\ 0\leq z\leq 1y,\ x^{2}\leq y\leq 1,\ 0\leq x\leq 1\}$
Hence the integral is
$\displaystyle {\begin{aligned}\iiint _{E}x\;{\text{d}}V&=\int _{0}^{1}\int _{x^{2}}^{1}\int _{0}^{1y}x\;{\text{d}}z{\text{d}}y{\text{d}}x\\&=\int _{0}^{1}\int _{x^{2}}^{1}x(1y)\;{\text{d}}y{\text{d}}x\\&=\int _{0}^{1}x\left[y{\frac {y^{2}}{2}}\right]_{x^{2}}^{1}\;{\text{d}}x\\&=\int _{0}^{1}{\frac {x}{2}}x^{3}+{\frac {x^{5}}{2}}\;{\text{d}}x\\=&\left[{\frac {x^{2}}{4}}{\frac {x^{4}}{4}}+{\frac {x^{6}}{12}}\right]_{0}^{1}={\frac {1}{12}}\end{aligned}}$

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MER QGH flag, MER QGQ flag, MER QGS flag, MER QGT flag, MER Tag Multiple integral, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag

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