Cupid's Arrows

This problem is quite mathematical, actually.

The magnetic flux is calculated by the following formula:
$\phi = B \cdot A$

With B given, there is a need to evaluate A .

Rearranging the given equation eventually leads to two equations:
$y = \dfrac{4}{5} (\sqrt{|x|} + \sqrt{1-x^2} )$ and $y = \dfrac{4}{5} (\sqrt{|x|} - \sqrt{1-x^2} )$

Evaluating the area via definite integrals,
$\displaystyle\int_{-1}^{1} \dfrac{4}{5} (\sqrt{|x|} + \sqrt{1-x^2} ) - \dfrac{4}{5} (\sqrt{|x|} - \sqrt{1-x^2} ) \text{d}x$

$\displaystyle\int_{-1}^{1} \dfrac{8}{5}( \sqrt{1-x^2}) \text{d}x\$

$\dfrac{8}{5} \cdot \dfrac{\pi}{2}$

$2.5$

Substituting back to the formula for magnetic flux results in $\boxed{2.5}$ Wb.

$\Phi = B_\perp A$ , and with $B_\perp = 1\ \text{Wb/m}^2$ this problem is equivalent to asking for the area.

The trick is to ignore the $\sqrt{|x|}$ term. Without it, the equation describes an ellipse. With the term, each vertical "strip" of the area is shifted upward without affecting its length.

So I look at $x^2 + (\tfrac 54 y)^2 = 1$ and recognize is as the equation of an ellipse. Its extreme points are when $x^2 = 1$ and $(\tfrac54 y)^2 = 1$ , so it follows that the semimajor and semiminor axes have lengths $a = 1$ and $b = \tfrac45$ respectively.

The area of an ellipse is $A = ab\pi = 1\cdot \tfrac 45\cdot \pi \approx \boxed{2.513}$ .