Need help solving a surface integral question

Question

The question is: I understand the method used in the book. However, when I try a slightly different method, I end up with an incorrect answer: 2$\pi$ instead of 4$\pi$ as given in the book. I have already tried looking for what I could be going wrong but no avail. Hence decided to ask here. Thanks in advance.

Answer 1

So basically you want to calculate $\int\int_S dA$. So half of it will be given by the parametrization $(x, y) \mapsto (x, y, \sqrt{a^2 - x^2 - y^2})$, for $x^2 + y^2 \le a^2$.

Then the area element is given by

$$\frac{a}{\sqrt{a^2 - x^2 - y^2}}dxdy = \frac{ar}{\sqrt{a^2 - r^2}} dr d\theta$$

So

$$ \int \int_S dA = \frac{2}{a} \int_{x^2 + y^2 \le a^2}\frac{dxdy}{\sqrt{a^2 - x^2 - y^2}}$$

That is, you only forgot the 2 there.

Answer 2

When you wrote out an expression for $dS$, it appears that you're only integrating over the top half of the sphere.