# Evaluate the integral over the helicoid [Surface integrals]

## Homework Statement

Evaluate the integral $\int\int_S \sqrt{1+x^{2}+y^{2}}dS$ where S:{ r(u,v) = (ucos(v),usin(v),v) |$0\leq u\leq 4,0\leq v\leq 4\pi$ }

2. The attempt at a solution

Here is my attempt, I am fairly sure I am right, but it is an online assignment and it keeps telling me I am wrong. I just wanted to double check before I contact the professor to see if he made a mistake.

Let x=ucos(v),y=usin(v) and,

the jacobian determinant is $u(cos^{2}(v)+sin^{2}(v))=u$

now my new integral is

$\int_0^{4\pi} \int_0^4 u\sqrt{(1+u^{2})} dudv$

Now this is a fairly straightforward problem and I do a simple substitution to get

let $\phi=1+u^{2},du=2u$

$\frac{1}{2}\int_0^{4\pi} \int \sqrt{\phi} d\phi dv=2\pi[\frac{2}{3}\phi^{\frac{3}{2}}]=2\pi[\frac{2}{3}(1+u^{2})^{\frac{3}{2}}]_0^4$

and finally after plugging in the values, I get

$2\pi(\frac{2}{3} 17^{\frac{3}{2}} - \frac{2}{3})$

Does anyone else see a flaw in this? again, I am pretty sure I am right but would appreciate it immensely if someone could point out my mistake!

Thank you

EDIT: Also if I made any typos in the equations my apologies, this is my first time editing with latex commands

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