What am I misunderstanding about this basic integral question? (Area under a curve) I'm confused about part B of the following exercise on basic integrals. I am asked to find the area of R2, the blue shaded area.
My method was to find the area of the area of the triangle (0,7), (0,0), (7,0) using $1/2bh$ or integration, then to subtract the area of the parabola between $x = 0$ and $x = 2$ (intersection) and the line.

However the printed solution is quite different and I don't understand why it is correct and my reasoning is not. They seem to have taken the area of the smaller triangle between $x = 2$ (right-most intersection of the line and the parabola) and $x = 7$, then added the area between the parabola, line and positive y-axis.
But doesn't that miss the part of R2 (blue area) between $x=2$ and the y-axis, below the parabola?

Thanks very much for your help!
p.s. Another quick, related question: Am I correct in thinking that in order to find the area under a curve, you only need to split the integral into two sums if the curve goes below the x-axis?

The integral $\int_0^2 (x^2+1) dx$ that they add to the triangle area is exactly the area of that part that you're worrying that they are missing.
