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Find the volume of the solid region in the first octant bounded by the coordinate planes, the plane $y + z = 2$ and the parabolic cylinder $x = 4 - y^2$.

I have a final answer, I would just like to make sure I am correct.

Because it is the first octant bounded by the coordinate planes: $x \geq 0, y \geq 0, z \geq 0$. This means that we can define the boundaries of the region as $0 \leq z \leq 2 - y, 0 \leq y \leq \sqrt{4 - x}, 0 \leq x \leq 4$ (from substituting $y = 0$ into $x = 4 - y^2$).

So E = { $(x,y,z) | 0 \leq x \leq 4, 0 \leq y \leq \sqrt{4 - x}, 0 \leq z \leq 2-y$} and the projected xy plane would be the function $y = \sqrt{4 - x}$.

V = $\int_0^4 \int_0^{\sqrt{4 - x}} \int_o^{2-y}dzdydx$ (I wanted to make sure this part was accurate in terms of order and logic).

V = $\int_0^4 \int_0^{\sqrt{4 - x}} (2-y)dydx$

V = $\int_0^4 [2 \sqrt{4-x} - \frac{4 - x}{2}]dx$

V = $2\int_0^4 \sqrt{4-x}dx - \frac{1}{2}\int_0^4(4 - x)dx$

V = $6\frac{2}{3}$ $units^3$

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  • $\begingroup$ Good, that makes much more sense. $\endgroup$ – David K Feb 11 '16 at 19:32
  • $\begingroup$ A couple things you can do to enhance readability: use \leq and \geq instead of <= and >=. I also recommend, when you have a long math expression, put the entire expression inside the delimiters $...$ (or even $$...$$) rather than just putting bits and pieces of the math inside $...$. To get { } inside $...$, use \{ and \}. There's more about formatting at meta.math.stackexchange.com/questions/5020/… (Good job using as much formatting as you did; it helped.) $\endgroup$ – David K Feb 11 '16 at 19:38
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The solution appears to be correct. Personally, I used different construction of the integral, which is

$\int_0^2dy\int_0^{2-y}dz\int_0^{4-y^2}dx = \frac{20}{3}$.

Hope this helps.

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  • $\begingroup$ I really appreciate the input! I guess the way I did it was just a few extra steps, and leaving the original functions in their natural form would have been a cleaner way to evaluate. Thanks for the help! $\endgroup$ – SmithNineSix Feb 13 '16 at 2:18
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Find the volume of the region in the first octant bounded by the coordinate planes, the plane $𝑥 + 𝑦 = 2$, and the cylinder $𝑦 ^2 + 4𝑧^ 2 = 16$.

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