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I need to create a VOLUME under f[x,y]= x (y^3 + 1)^(1/2) that's above the triangular region bounded by y=x/3, y=2, and x=0. I know how to make the plot of the function over an area, such as Plot3D[f[x, y], {x, 0, 6}, {y, 0, 2}]. However, I do not know how to make it exclusively over the triangle.

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f[x_, y_] = x (y^3 + 1)^(1/2);

Plot3D[f[x, y], {x, 0, 6}, {y, x/3, 2}, Filling -> 0]

enter image description here

Plot3D[f[x, y], {x, 0, 6}, {y, 0, 2}, 
 RegionFunction -> Function[{x, y}, x/3 <= y <= 2 && x >= 0], 
 Filling -> 0]

enter image description here

RegionPlot3D[0 <= z <= f[x, y] && x/3 <= y <= 2 && 0 <= x <= 6,
 {x, 0, 6}, {y, 0, 2}, {z, 0, 18}, PlotPoints -> 51]

enter image description here

Maximize[{f[x, y], x >= 0, x/3 <= y <= 2}, {x, y}]

{18, {x -> 6, y -> 2}}

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Several methods. one of them is:

Plot3D[x (y^3 + 1)^(1/2), {x, 0, 6}, {y, 0, 2}, 
 RegionFunction -> Function[{x, y}, y <= 2 && y >= 3 x && x >= 0]]
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  • $\begingroup$ Great catch. thanks :) $\endgroup$ – Basheer Algohi Nov 24 '14 at 4:01
  • $\begingroup$ You're welcome! (+1 already.) $\endgroup$ – Michael E2 Nov 24 '14 at 4:01
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In V10+:

Plot3D[x (y^3 + 1)^(1/2),
 {x, y} ∈ Polygon[{{0, 0}, {6, 2}, {0, 2}}],
 AxesLabel -> Automatic, Filling -> 0]

Also

DiscretizeRegion[
 ImplicitRegion[0 <= z <= x (y^3 + 1)^(1/2), {{x, 0, 6}, {y, 0, 2}, {z, 0, 18}}]
 ]
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  • $\begingroup$ I tried this and realized I wrote the problem incorrectly. I don't need the slice of f[x,y] over the triangle. Instead I need the volume between the triangle on the xy plane and f[x,y]. It should be a solid with a triangular base. Thank you for your quick response, and I apologize for my poor wording. This is my first time using the program. $\endgroup$ – Nick Nov 24 '14 at 4:06
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NIntegrate x (y^3 + 1)^(1/2) over Polygon[{{0, 0}, {6, 2}, {0, 2}}]:

NIntegrate [x (y^3 + 1)^(1/2), {x, y} \[Element] Polygon[{{0, 0}, {6, 2}, {0, 2}}]]

Answer:26.

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