What is the area of the surface between a curve and its projection onto a plane? $ \int_Cf(x,y)\,d\boldsymbol{s}=\int_a^bf\big(x(t),y(t)\big)\sqrt{\Big(\frac{dx}{dt}\Big)^2+\Big(\frac{dy}{dt}\Big)^2}dt $ - Parameterize $x,y$ in terms of new variable $t$