Find the area between curves using definite integration, accounting for intersection points and which curve is on top

What this dot point is asking

SCSA wants you to set up and evaluate the area between two curves, locate the boundaries by solving for intersections, and handle regions where the upper and lower curves change.

Top minus bottom

The integrand is the vertical gap between the curves at each $x$. Because we subtract the lower curve, the result is positive and there is no need to worry about whether either curve dips below the $x$-axis: the difference handles that automatically.

Finding the limits

The limits $a$ and $b$ are usually the $x$-coordinates of the intersection points, found by solving $f(x) = g(x)$. Sketch the region to confirm which curve is on top, since that determines the order of subtraction.

When the curves cross inside

Integrating with respect to y

When the region is bounded more naturally on the left and right, it can be cleaner to write $x$ as a function of $y$ and integrate $\int_c^d [x_{\text{right}} - x_{\text{left}}]\,dy$. Choose the variable that makes the bounds simplest.