How do I construct the graph of the solid limited by: $\displaystyle 2y^2=x$ , $\displaystyle \frac{x}{4}+\frac{y}{2}+\frac{z}{4}=1$ , $\displaystyle z=0$ , $\displaystyle y=0$

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- Nov 13th 2008, 07:44 AMApprentice123graphicHow do I construct the graph of the solid limited by: $\displaystyle 2y^2=x$ , $\displaystyle \frac{x}{4}+\frac{y}{2}+\frac{z}{4}=1$ , $\displaystyle z=0$ , $\displaystyle y=0$
- Nov 13th 2008, 09:01 AMshawsend
Hello Apprentice. I don't know how you guys do those triple integrals without plotting them first. Actually I had this when those guys were helping you with it a few days ago but thought it wasn't necessary. It's a little tough to see without interactively rotating it but below is the domain of integration and the other is the three surfaces. The volume is under the blue section. Look carefully at the domain and then it's easy to see that its:

$\displaystyle \int_0^1\int_{2y^2}^{4-2y}\int_0^{4-x-2y} dzdxdy$

Here's the Mathematica code to draw the surfaces. Also, don't get discouraged by the code. Once you get good at it, it only takes a few minutes to code it.

Code:`polys = Graphics3D[{Opacity[0.2],`

LightPurple, {Polygon[{{5, 0, 5},

{-5, 0, 5}, {-5, 0, -5},

{5, 0, -5}}]}}];

polys2 = Graphics3D[{Opacity[0.8],

LightPurple, {Polygon[{{5, 5, 0},

{-5, 5, 0}, {-5, -5, 0},

{5, -5, 0}}]}}];

c1 = ContourPlot3D[{2*y^2 == x}, {x, 0, 5},

{y, -2, 2}, {z, 0, 5}]

p1 = Plot3D[4*(1 - x/4 - y/2), {x, 0, 4},

{y, 0, 2}, PlotStyle -> {Opacity[0.5]}]

p2 = Plot3D[4*(1 - x/4 - y/2), {x, 0, 4},

{y, 0, 2}, PlotStyle -> Blue,

RegionFunction -> Function[{x, y},

x > 2*y^2 && 4*(1 - x/4 - y/2) > 0]]

final = Show[{p1, p2, polys, polys2, c1},

BoxRatios -> {1, 1, 1}, AxesLabel ->

{Style["X", 20], Style["Y", 20],

Style["Z", 20]}]

domain = Plot[{Sqrt[x/2], 2 - x/2},

{x, 0, 5}]

GraphicsGrid[{{domain, final}}]

- Nov 13th 2008, 09:39 AMApprentice123
- Nov 13th 2008, 10:12 AMshawsend
Wait a minute Apprentice . . . that code has nothing to do with the integral. I'm just plotting the graphs without even thinking about the integral, and then looking at the graphs to figure out the limits of an integration. I hope I'm not causing confusion by posting complicated code which interferes with the underlying mathematics.

Just plot one thing:

Code:`p2 = Plot3D[4*(1 - x/4 - y/2), {x, 0, 4},`

{y, 0, 2}]