Find the area of the part of the cylinder $\displaystyle x^2+y^2=1$ that is given by $\displaystyle 0<= z <= 2-x^2-2y^2$.

(Hint: parametrize the surface.)

I did manage to calculate the correct area by doing the following:

Cylinder parametrization in xy-plane:r(t)=(x(t), y(t)) with x(t)=cos(t), y(t) = sin(t)

Insert x(t) and y(t) in paraboloid equation to get area = $\displaystyle \int\limits_0^{2\pi } {(2 - \cos ^2 t - 2\sin ^2 t)dt}

$ (ds of r(t) is 1*dt)

How would you calculate the area by parametrizing the surface? For starters, how do you parametrize the surface?