How can i understand which take out from which when im calculating volume by integration

Question

Here is the question i deal with. In the solution it takes out one from another in step 2. (It does in in blue text) Why $3 \gt y^2+1$?

Answer 1

Because we are interested in the volume of revolution of the portion of the parabola $x = y^2+1$ that is to the "left of" the line $x = 3$ (and so the value of $x$ on the line $x = 3$ is greater than or equal to the value of $x$ on the parabola in the portion of the graph we are rotating.

So we need to consider the radius of the area bounded between the parabola (on the left) and the line $x = 3$ (on the right), and so the need to subtract $x= y^2 + 1$ from $x = 3$.

That gives us $R(y) = \underbrace{3}_{x = 3} - \underbrace{(y^2 + 1)}_{x = y^2 + 1}$

Note that you can view this area in the following graph. Wolfram Alpha put graphed this with $y$ measured horizontally, but you can tilt the grap 90 degrees clockwise to see the bounded region as it would be seen in a graph with the customary position of the x, y axes.