How to find the area of the region, bounded by various curves? 
Find the area of the region bounded by the curves $y=x^2$ and $y=x$.
Find the area of the region bounded by the curves $y=x^2+1$ and $y=2$

I have a ton of questions like this and I have been graphing them and then splitting them into intervals and adding them up but this is giving me an answer thats a little off and its taking forever....is there a faster way? Also I am stuck on $y=x^2+1$ and $y=2$
because I dont know what region they want..I see $y=2$ as a line intersecting $x^2+1$, when I graph it.

HINT They ask for the area of the yellow region:

The areas would be given by integrals $\int_{x_1}^{x_2} \left(y_\text{top}(x) - y_\text{bottom}(x)\right) \mathrm{d} x$ with appropriate choices of boundaries $x_1$ and $x_2$ and functions $y_\text{top}(x)$ and $y_\text{bottom}(x)$.
