A “hill” made from an upside paraboloid, and its surface area.

Define the hill as a function for easy access later, and plot it just to see what it looks like:

h[t_,Θ_] := { t * Cos[Θ], t * Sin[Θ], 1-t^2 }

ParametricPlot3D[ h[t,a], {t,0,1}, {a,0,2*Pi} ]

Start finding the surface area of the hill by getting its derivatives with respect to each parameter.

dHdT = D[ h[t,a], t ]

dHdTheta = D[ h[t,a], a ]

Next we need the cross product of the derivatives, i.e., the normal to the surface. Simplify this where there’s an opportunity, to reduce chances for problems later.

normal = Cross[ dHdT, dHdTheta ]

normal = Simplify[ normal ]

What we’ll actually want to integrate is the magnitude of the normal. This time simplifying by hand is helpful.

mag = Norm[ normal]

Simplify[ mag ]

mag = t * Sqrt[ 1 + 4t^2]

Finally, integrate the magnitude of the cross product of the derivatives. This can be either numeric or symbolic, whichever you prefer.

NIntegrate[ mag, {t,0,1}, {a,0,2*Pi} ]

Integrate[ mag, {t,0,1}, {a,0,2*Pi} ]

Finally, Mathematica has a built-in “Area” function that will do all of these calculations in one command.

Area[ h[t,a], {t,0,1}, {a,0,2*Pi} ]

N[ Area[ h[t,a], {t,0,1}, {a,0,2*Pi} ] ]