I am trying to compute an approximate value for the integral S given by r[x,y} = {x,y,xy} on
0≤x≤y,0≤y≤1
: →
∫
s
∫x+
2
y
+
3
z
S

Calculate Transformation formula using
3
z
= (x
3
y)
In[]:=
Simplify[Norm[(D[{
3
(xy)
},x])+(D[{
3
(xy)
},y])],Assumptions{0≤x≤y,0≤y≤1}]
Out[]=
3
2
x
2
y
(x+y)
Using the expansion of
3
(xy)
→
3
x
3
y
and transformation value calculated above in NIntegrate function to calculate surface area.
In[]:=
NIntegrate(x+
2
y
+
3
x
3
y
)
3
2
x
2
y
(x+y)+1
,{y,0,1},{x,0,y}
Out[]=
0.605618
Using SurfaceIntegrate to calculate surface area.
In[]:=
SurfaceIntegrate[(x+
2
y
+
3
z
),{x,y,z}∈ParametricRegion[{x,y,xy},{{x,0,y},{y,0,1}}]]//N
Out[]=
0.617639
The values generated by the two methods are close, but not exact. I believe the correct answer is 0.62 generated by the SurfaceIntegrate function. However because of the difference, I am concerned that I may be doing something incorrectly with the NIntegrate function and would appreciate any advice you could give.
​
​In using the Integrate function below, it ran forever but eventually gave the same answer as NIntegrate. Again probably because I am doing something incorrectly, and I would appreciate your input.
In[]:=
1
∫
0
y
∫
0
(x+
2
y
+
3
x
3
y
)
3
2
x
2
y
(x+y)+1
xy//N
Out[]=
0.605618
Plotting - Interesting that I can use ParametricPlot function to generate regions of both the parametric representation and the representation of the scalar formula. I did try Plot3D but it was difficult to understand since it appeared to plot (x, y, z ) separately.
In[]:=
ParametricPlot3D[{x,y,xy},{y,0,1},{x,0,y}]
Out[]=
In[]:=
ParametricPlot3D[{x,
2
y
,
3
(xy)
},{y,0,1},{x,0,y}]
Out[]=
In[]:=
Plot3D[{x,
2
y
,
3
(xy)
},{y,0,1},{x,0,y}]
Out[]=