Question

Find the area of the surface correct to four decimal places by expressing the area in terms of a single integral and using your calculator to estimate the integral. The part of the surface z=cos(x^2+y^2) that lies inside the cylinder x^2+y^2=1

Answer 1

If we substitute $x=r\cos\theta$ and $y=r\sin\theta$, we get $r^2=x^2+y^2$, so that

$z=\cos(x^2+y^2)=\cos(r^2)$

which is independent of $\theta$, which in turn means the surface can be treated like a surface of revolution.

Consider the function $f(t)=\cos(t^2)$ defined over $0\le t\le1$. Revolve the curve $C$ described by $f(t)$ about the line $t=0$. The area of the surface obtained in this way is then

$\displaystyle2\pi\int_C\mathrm dS=2\pi\int_0^1\sqrt{1+f'(t)^2}\,\mathrm dt$

$=\displaystyle2\pi\int_0^1\sqrt{1+(-2t\sin(t^2))^2}\,\mathrm dt$

$=\displaystyle2\pi\int_0^1\sqrt{1+4t^2\sin^2(t^2)}\,\mathrm dt\approx7.4144$