Find the area between the curves $y=\ln(x), y=1, y=-1, y^2 = x+2$
Doing a sketch on Desmos, I see this graph
The beige coloring is the area in between the graphs
It is obvious I have to break up this graph and calculate their respective areas seperately, and then add. How do I know where to break up though?
This is where I have decided to break up the graphs and calculate their area and add it all up. Is this a reasonable breaking structure, or am I doing too much work?
What you want to do is integrate with respect to $y$. Turn the graph in your head if you need to. What you get is $$\int _{-1}^1\left(e^y-\left(y^2-2\right)\right)dy$$
As you noted yourself in the comments. To evaluate this integral we can split into three separate integrals; the first is trivial because $\frac{d}{dx} e^x = e^x$, and the second/third require the power rule of integration, i.e. $\int x^n = \frac{x^{n+1}}{n+1}$