Question: SurfaceInt VectorCalc and plots


This is probably not very much maple related question, but to some extend it is.

After failing this question on my exam I have tried to solve it, but it seems like I cant get it right.

Given a sphere z^2+r^2=4 and a cylinder r=1 I was told to set up the volume integral for the element T enclose byt the outer sphere and the inner circle.

I tried to generate a plot but my skills are rather poor in plotting, if I could get the plot right I would be able to set up the volume integral. I have also tried to figure out how to do the surface integral and chose to use a task template as it is a bit more convinient when you find the syntax hard.

I would say I am familiar with the VectorCalculus:-SurfaceInt in cartesian for when i have intersection of two surfaces given in terms of z=


but this kind of problem is new to me.



Surface Integration over a Surface Defined Parametrically


Formulate and evaluate the surface integral of f(x, y, z) over a surface defined parametrically.



Surface Integral on a Surface Defined Parametrically









   " x(u,v)="










   " z(u,v)="







`≡`(F(u, v), f(x(u, v), y(u, v), z(u, v)))

`≡`(LinearAlgebra[Norm](N), sqrt((`∂`(y, z)/`∂`(u, v))^2+(`∂`(z, x)/`∂`(u, v))^2+(`∂`(x, y)/`∂`(u, v))^2))




"∫(∫)[S]f ⅆsigma =(∫)[u=a]^(u=b)(∫)[v=g(u)]^(v=h(u))F(u,v)||N|| ⅆv ⅆu"



"∫(∫)[S]f ⅆsigma=""(∫)[v=a]^(v=b)(∫)[u=G(v)]^(u=H(v))F(u,v)||N|| ⅆu ⅆv"
















p1 := plot3d([2*cos(u)*sin(v), 2*sin(u)*sin(v), 2*cos(v)], u = 0 .. 2*Pi, v = 0 .. Pi, color = green, transparency = .55):

p2 := plot3d([cos(u), sin(u), z], u = 0 .. 2*Pi, z = -2 .. 2, color = red, filled = true):

display(p1, p2)


p3 := plots:-sphereplot(2, theta = 0 .. 2*Pi, phi = 0 .. Pi, color = green, transparency = .55):

p4 := plots:-cylinderplot(1, theta = 0 .. 2*Pi, z = -2 .. 2, color = red):

plots:-display(p1, p2)






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