I am given this question below, and I was hoping for some guidance as to why I am getting my answer incorrect...
Question:
Consider the flux integral
∬Sf⋅n̂ dS, where f= [0,y,2z] and S is the paraboloid surface shown below.
This paraboloid has height 2 and maximum radius 2.
(a) Based on cylindrical polar coordinates, enter the parameterisation
p(r,t)=(x(r,t),y(r,t),z(r,t)), such that 0≤r<2 and 0≤t<2π gives the paraboloid surface above.
So, my answer is: [rcos(t), rsin(t), 0.5*r^2*(cos(t)^2) + 0.5*r^2*(sin(t))^2], and when I plotted this, my graph seems to be identical to the above graph, yet my answer is wrong. My teacher said that I need to consider the height with the paraboloid when the radius is 2, and that I am only a constant out...
I appreciate any help!
The paraboloid can be expressed as
$$x^2+y^2=2z$$
which can be parametrized as below,
$$x(r,t)=r\cos t$$ $$y(r,t)=r\sin t$$ $$z(r,t)=\frac 12 r^2$$
You may just have to simplify your $z$ function.