Faraday's law with Calc 3 integration help

[horsewnoname]

Homework Statement

Suppose an infinitely long wire carrying current $I=sin_0(\omega t)$ is a distance $a$ away from a equilateral triangular circuit with resistance $R$ in the same plane as shown in the figure. Each side of the circuit is length $b$. I need to find the induced voltage $V_{ind}$ and current $i$ of the triangular circuit.

Homework Equations

Faraday's law $V_{ind}=-\frac{\mathrm{d} \phi}{\mathrm{d} x}$ where magnetic flux $\phi = \int \boldsymbol{B}\cdot d\boldsymbol{s}$ and $d\boldsymbol{s}$ is the surface through which the flux intercepts.

Also relevant is Ohm's law which is simply $V=IR$.

The Attempt at a Solution

So the first thing I have to do is determine the $\boldsymbol{B}$ produced by the infinite wire. This is simply $\boldsymbol{B}=\frac{\mu_0 I}{2 \pi x}$.

From this, I can determine the magnetic flux which is the tricky part. To do this, I need to integrate over the area of the triangle which means I need a double integral where the horizontal $x$ limits of integration are given by $x=a$ to $x=a+\frac{\sqrt3}{2}b$ and the vertical $y$ limits of integration are given by $y=0$ to $y=\frac{-1}{\sqrt3}x+\frac{a}{\sqrt3}+\frac{b}{2}$. I got this equation of a line by splitting the triangle in half as I show in the figure above. The only thing is that I will have to multiply the result by 2 to consider both areas of the triangle, not just half.

It's been a while since I've taken Calc 3, but I eventually got to the result $\frac{\mu_0I}{\pi} [\frac{-b}{2}+(\frac{a}{\sqrt3}+\frac{b}{2}) ln(\frac{a+\frac{\sqrt3 b}{2}}{a})]$.

From here, I know I can just plug $\phi$ into Faraday's law and get the induced voltage, but I don't know if I did the integral correctly. Is there any way for me to check that I have the correct result?

 

[TSny]

A partial check to see if your expression for y and your limits are correct is to see if you get the correct area for the triangle if you evaluate ∫ydx.

Your explanation of how you set up the integral sounds right. I think your answer for the flux is correct. (Your parentheses look a little out of place.)