Area of a sheet: Interpreting line integrals with respect to arc length.
> | 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
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"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
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In other words, and
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"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; |
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(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)); |
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Example: Plot and find the net area of the sheet above/below and the ellipse
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> | 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; |
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(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); |
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