Question

Surface Area Integral

The question is below, thanks for any help.

Answer 1

Integrate the following:
\[
\int_0^1 dx\int_0^{\sqrt{1-x^2}} dy \sqrt{1+4x^2+4y^2}
\]

Where the shape in question is a paraboloid.

Answer 2

@satellite73

Answer 3

Never mind, the only nice way to do this is to use cylindrical coordinates... Egh.

Answer 4

Just for further reference:
\[
\int_\gamma dx\,dy \sqrt{1+4x^2+4y^2}=\int_Sd\theta\,dr\,r \sqrt{1+4r^2}=2\int_{[0,2\pi]}d\theta \int_{[0,4]}\sqrt{1+u}=\frac{\pi}{3}(5\sqrt{5}-1)
\]

Answer 5

\[\begin{align}
\int_\gamma dx\,dy \sqrt{1+4x^2+4y^2}&=
\\
\int_Sd\theta\,dr\,r \sqrt{1+4r^2}&=
\\
2\int_{[0,2\pi]}d\theta \int_{[0,4]}\sqrt{1+u}&=\\&=\frac{\pi}{3}(5\sqrt{5}-1)
\end{align}
\]