# Find the area between three curves

I've been given three curves $y=5+\sin(x)$, $y=2x+1$ and $y=-x+0.5$. I need to find the area between these curves. Usually I need to find the upper and lower bounds first and then integrate, but there is no way to algebraically solve equations like $x+\sin(x)=4.5$. I also thought area formula in polar coordinates, but I couldn't find a way to convert $y=5+\sin(x)$ into polar form. Should I just approximate the bounds or is there a better way?

I would just approximate the intersections as you suggest. Certainly integrating in Cartesian coordinates is the way to go. I am a fan of fixed point iteration for finding intersections like this because I find it quicker to set up than other root finders so it minimizes the person plus computer time. We can say we want to solve $x+\sin x = -4.5$. Since the derivative of $\sin$ is always less (or equal to) than $1$ in absolute value we can just write $x=-4.5-\sin x$ which converges to $x \approx -5.3206$ and $x=\frac 12(4+\sin x)$ which converges to $x \approx 2.3542$