Find the surface area of cone $ {x^2 + y^2 = z^2} $ cut off by surface of cylinder $ {x^2 + y^2 = a^2} $ above the $xy$ plane.
My approach:
I considered projection of the area on $xy$ plane cut off by cylinder as the circle C: $ {x^2 + y^2 = a^2}, x\ge 0, y\ge 0 $.
Surface area = $4\,\,\int\int\sqrt{1 + (\partial{z/\partial{x}})^2 + (\partial{z/\partial{y}})^2} dxdy$ over the circle C.
Converting it to polar form by using $x = rcosθ, y = rsinθ$ gives limits $r: 0\to a$ and $θ: 0\to π/2$.
After integration, I got the final value of area as $\sqrt2π{a^2}$ but the answer in the book is given as $2π{a^2}$.