Calculate the area enclosed between two curves using definite integration.

What this dot point is asking

You already know how to find the area between a single curve and the $x$-axis. This dot point extends that to the region trapped between two graphs, which is one of the most common extended-response integration questions.

The core idea

If $f(x) \ge g(x)$ over an interval, the vertical gap between the curves at each $x$ is $f(x) - g(x)$. Summing those gaps across the interval by integration gives the enclosed area.

Subtracting before integrating means the result is correct even when part of the region lies below the $x$-axis, because the two unwanted areas cancel and only the gap between the curves remains. This is the big advantage over computing each area separately.

Finding the limits

The limits are almost never given directly; you find them by setting the curves equal.

Deciding which curve is on top

Test a single point between the limits. Substitute an $x$-value from inside the interval into both functions; whichever gives the larger $y$ is the upper curve $f$. Getting this backwards produces a negative answer, which is a clear signal you have swapped them.

When the curves cross inside the region

If the two curves intersect at a point between the outer limits, the upper and lower curves swap there. In that case split the integral at the crossing and write each piece as upper minus lower for that sub-interval, then add the absolute areas. Skipping the split lets the two parts partly cancel and gives too small an answer.

Summary

Set the curves equal to find the intersection limits, decide which curve is upper by testing a point inside, then integrate upper minus lower across the region. If the curves cross within the interval, split at that point and add the absolute areas. Subtracting the functions before integrating automatically handles regions below the axis, which is why this method is cleaner than computing separate areas.