"Find the area of the portion of the elliptic paraboloid $ \mathbb z =x^2/2a + y^2/2b $ that is inside the cylinder $ \mathbb x^2 / a^2 + y^2 / b^2 = c^2 $ Hint : Parametrise the elliptic paraboloid as $ \mathbb x= au , y=bu , z= au^2 /2 + bv^2 / 2 $ "
So I got took partial derivatives and got the norm of the cross product of these tangent vectors to be $ \mathbb ab\sqrt(u^2 + v^2 +1) $
So now have that the surface area is = $ \mathbb \int_{-b}^b \int_{-a}^a ab\sqrt{u^2 + v^2 +1} du dv$
now I think polar coordinates are best way the compute this but not sure how I can change the ellipse projection on the $ u v$ plane. Now I'm not actually sure if I even set the whole thing up right. It's my first time working with parametrised surfaces so am still unsure how everything comes together especially changing the parameters to another coordinate system to make integration easier.
Can anyone tell me if I've done this right so far and how I could proceed in computing this area?
any help or advice is much appreciated