Calculate the Area of Toroidal

Question

i need help with the integral of this function.

$ f(x,y)=\left( x-b \right) ^{2}+{y}^{2}={a}^{2}$

Note: $0 < a < b$

I know the graphics of this function is this http://www.maplesoft.com/support/help/Maple/view.aspx?path=MathApps/ToroidalCoordinates

but i cannot put the double integral, can help me?

Answer 1

Let $f(\varphi, \theta) = ((b + a \cos\theta) \cos(\varphi),~ (b + a\cos\theta) \sin(\varphi),~ a\sin\theta)$.

Then the torus $T$ with major radius $b$ and minor radius $a$ is given parametrically by $T = f([0,2\pi) \times [0, 2\pi))$.

Now the surface integral of $T$ is, as always, defined by parametrizations as $$\oint_T 1ds = \int_0^{2\pi}\int_0^{2\pi}1\|f_\varphi(\varphi, \theta) \times f_\theta(\varphi,\theta)\| ~d\theta ~d\varphi,$$ where $f_\varphi$ and $f_\theta$ are the derivatives of $f$ along the first and second parameter. With this formula, you should be able to calculate the area of $T$ given $a$ and $b$.