Area of a sheet: Interpreting line integrals with respect to arc length. 

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restart; with(plots): with(VectorCalculus): sheetPlot := proc(f,fDom,r,rDom)   local plotSurface, planeCurve, surfaceCurve, verticalSurface, plotTitle;   plotSurface := plot3d(f(x,y),x=fDom[1],y=fDom[2],scaling=constrained,transparency=0.75);   planeCurve := spacecurve(<r(t).<1,0>,r(t).<0,1>,0>,t=rDom,thickness=3,color=black);   surfaceCurve := spacecurve(<r(t).<1,0>,r(t).<0,1>,f(r(t).<1,0>,r(t).<0,1>)>,t=rDom,thickness=2,color=blue,numpoints=5000);   verticalSurface := plot3d(<r(t).<1,0>,r(t).<0,1>,s*f(r(t).<1,0>,r(t).<0,1>)>,t=rDom,s=0..1,color=green,numpoints=10000);   #plotTitle := cat("Area = ",int(f(r(t).<1,0>,r(t).<0,1>)*Norm(r(t)),t=rDom)," = ",evalf(int(f(r(t).<1,0>,r(t).<0,1>)*Norm(r(t)),t=rDom)));   #display({plotSurface,planeCurve,surfaceCurve,verticalSurface},scaling=constrained,title=plotTitle);   # No plot title.   display({plotSurface,planeCurve,surfaceCurve,verticalSurface},scaling=constrained); end proc:

 

Example: Let plot and compute the area of the sheet below and above the half circle where . 

"fDom" specifies the plot domain for the function . Let's plot the graph of this function when its domain is restricted to the interior of the disk .  

In other words, and . 

"rDom" specifies the domain for our parameterized curve. In this case we parameterize the circle (using cosine and sine) and get . 

To restrict ourselves to the upper half of this circle we need . 

>	f := (x,y) -> 4-x^2-y^2: 'f(x,y)' = f(x,y); fDom := [-3..3,-sqrt(9-x^2)..sqrt(9-x^2)]; r := t -> <cos(t)+1,sin(t)>: 'r(t)' = r(t); rDom := 0..Pi;

 

 

 

 

	(1)

 

The "Int" command is inert. This is just there to print out a "nice" looking line integral. [Note:  \x20 is the code for a blank space. This makes the upper bound in the integral appear empty.] 

I added in a yellow xy-plane. Recall that the line integral is computing net area under the curve just like a regular definite integral. 

>	Int(f(x,y),s=C..\x20) = int(f(r(t).<1,0>,r(t).<0,1>)*Norm(diff(r(t),t)),t=rDom); display(sheetPlot(f,fDom,r,rDom),plot3d(0,x=-3..3,y=-3..3,color=yellow));

 

Example: Plot and find the net area of the sheet above/below and the ellipse . 

>	f := (x,y) -> 2*sin(x^2+y): 'f(x,y)' = f(x,y); fDom := [-Pi..Pi,-Pi..Pi]; r := t -> <3*cos(t),2*sin(t)>: 'r(t)' = r(t); rDom := 0..2*Pi;

 

 

 

 

	(2)

 

>	Int(f(x,y),s=C..\x20) = int(f(r(t).<1,0>,r(t).<0,1>)*Norm(diff(r(t),t)),t=rDom); Int(f(x,y),s=C..\x20) = evalf(int(f(r(t).<1,0>,r(t).<0,1>)*Norm(diff(r(t),t)),t=rDom)); display(sheetPlot(f,fDom,r,rDom),viewpoint=circleleft);