Calculate area of subsection of a three-dimensional surface

[kwsockman]

For the following three-dimensional surface, z = -4.53 + 2.67x + 2.78y - 1.09xy, I would like to calculate the area for each of three subsections of this surface: (1) for which z is in between the corresponding x and y values (i.e., x < z < y OR y < z < x); (2) for which x is in between the corresponding z and y values (i.e., z < x < y OR y < x < z); (3) for which y is in between the corresponding z and x values (i.e., z < y < x OR x < y < z). Can anybody give me a formula for doing this so that I can do additional versions of this with different coefficient values?

If it helps to clarify and if you have Mathematica, I have plotted the surface in Mathematica and colored the regions indicated above.

f[x_, y_] = -4.53 + 2.67 x + 2.78 y - 1.09 x y // Rationalize // Simplify;

Plot3D[f[x, y], {x, 1.8, 2.6}, {y, 1.8, 2.6}, PlotRange -> {1.7, 2.6}, PlotPoints -> 101, AxesLabel -> (Style[#, Bold, 14] & /@ {"x", "y", "z"}), Ticks -> {{1.8, 2., 2.2, 2.4, 2.6}, {1.8, 2., 2.2, 2.4, 2.6}, {1.8, 2., 2.2, 2.4, 2.6}}, BoxRatios -> {1, 1, 1}, Mesh -> None, ColorFunction -> Function[{x, y, z}, Piecewise[{{Pink, (x < z < y) || (y < z < x)}, {Gray, (z < x < y) || (y < x < z)}}, Yellow]], ColorFunctionScaling -> False]

The three regions above are colored pink, gray, and yellow, respectively.

Thanks in advance,
Keith

 

[jvicens]

Dear Keith,

Thank you for your forum post. I am happy to help with your request for a formula to calculate the area for each of the three subsections of the given three-dimensional surface. The formula will depend on the specific values of the coefficients in the given equation, but I can provide a general approach that can be applied to different versions of the equation.

First, let's define the given equation as a function in Mathematica:

f[x_, y_] = -4.53 + 2.67 x + 2.78 y - 1.09 x y

Next, we can define the three subsections of the surface as separate functions, using the conditions given in the forum post. For example, for the first subsection where z is between the corresponding x and y values, we can define:

f1[x_, y_] := Piecewise[{{f[x, y], x < z < y || y < z < x}}, 0]

Similarly, for the second and third subsections, we can define:

f2[x_, y_] := Piecewise[{{f[x, y], z < x < y || y < x < z}}, 0]

f3[x_, y_] := Piecewise[{{f[x, y], z < y < x || x < y < z}}, 0]

Next, we can use the built-in Mathematica function "RegionPlot" to plot the three subsections and calculate their areas. For example, for the first subsection, we can use:

RegionPlot[f1[x, y] != 0, {x, 1.8, 2.6}, {y, 1.8, 2.6}, PlotPoints -> 100, PlotRange -> {{1.7, 2.6}, {1.7, 2.6}}, FrameLabel -> (Style[#, Bold, 14] & /@ {"x", "y"}), ImageSize -> Medium]

This will plot the first subsection as a shaded region and also calculate its area. The same can be done for the other two subsections.

Alternatively, if you want to calculate the areas directly without plotting, you can use the built-in Mathematica function "NIntegrate". For example, the area of the first subsection can be calculated as:

NIntegrate[f1[x, y], {x, 1.8, 2.6}, {y,