Pillowed Diamond

Because working with absolute values is hard, let us confine our working space to non-negative $x$ and $y$ . This way, we can get rid of the absolute values and find the area of one fourth of the pillowed diamond, then multiply by $4$ to find the total area.

Since $x,y > 0$ , our equation simplifies to $x+y-xy=0.5$ .

Factoring $y$ out gives $x+y(1-x)=0.5$ .

Now we can isolate $y$ and create a function of $x$ : $y=\dfrac{0.5-x}{1-x}$

This can be simplified to $\boxed{y=\dfrac{1}{2x-2}+1}$

Now that we have found our bounding equation for the top right side of the pillowed diamond, we can take the integral if the equation to ind the area underneath it.

First, we must find the upper and lower bound. The lower bound is obviously $0$ . The upper bound occurs when $y=0$ , which gives $|x|=0.5$ . Since we are only considering positive $x$ , the upper bound is $0.5$ .

Thus, we want to find the value of $\int_0^{0.5}\dfrac{1}{2x-2}+1\text{ d}x$

First, we can divide it into two sections: $\int_0^{0.5}1\text{ d}x+\int_0^{0.5}\dfrac{1}{2x-2}\text{ d}x$ which simplifies into $x+\dfrac{1}{2}\int_0^{0.5}\dfrac{1}{x-1}\text{ d}x$

Substituting $u=x-1$ (which means $\text{d}u=\text{d}x$ ) we simplify it into $x+\dfrac{1}{2}\int_1^{1.5}\dfrac{1}{u}\text{ d}u$

This is simply $x+\left.\dfrac{1}{2}(\ln |u|)\right|_1^{1.5}$ which becomes $x+\left.\dfrac{1}{2}(\ln|x-1|)\right|_0^{0.5}$

This evaluates as $\left(\dfrac{1}{2}+\dfrac{1}{2}\left(\ln \dfrac{1}{2}\right)\right)-\left(0+\dfrac{1}{2}(\ln 1)\right)=\dfrac{1}{2}-\dfrac{1}{2}\ln 2$

This is the area of a quadrant of the pillowed diamond. The area of the entire diamond is thus $4\left(\dfrac{1}{2}-\dfrac{1}{2}\ln 2\right)=2-2\ln 2\approx 0.613$ and our answer is $\boxed{613}$ .