Finding the bounds: the points of intersection

To compute an area between two curves, you still need to know between which x-values to integrate. Often, the bounds are not given: they are the points of intersection of the two curves, where the region closes up.

You find them by solving the equation that sets the two functions equal:

$$f(x) = g(x)$$

whose solutions give the bounds $a$ and $b$.

Example. Take $y = x$ and $y = x^2$ again. Their intersections:

$$x = x^2 \iff x^2 - x = 0 \iff x(x - 1) = 0 \iff x = 0 \text{ or } x = 1$$

The bounds of the area are therefore $a = 0$ and $b = 1$ — exactly the ones used in the previous lesson. The region is "closed off" by these two meeting points.

Watch out for crossings

If the curves cross inside the interval, the top curve and the bottom curve swap roles. You must then split at each intersection and, on each piece, subtract in the right order: top minus bottom.

The complete procedure comes down to three steps:

1. Solve $f(x) = g(x)$ to get the bounds and the crossings.
2. On each piece, identify the top curve.
3. Integrate the difference top minus bottom, then add up.

Remember that the hardest part is not the integral, it's the setup: finding the bounds and knowing which curve dominates which. Once that is settled, the computation follows on its own.