Finding the Area Between Curves Expressed as Functions of x

A single integral of f(x) gives area under a curve down to the x-axis. When you want the area trapped between two curves, that x-axis baseline is replaced by the second curve. At every x value, the region has a height equal to the top function minus the bottom function. Stack up all those tiny vertical slivers, each with width dx and height top minus bottom, and the integral sweeps them across the interval, giving the exact trapped area.

The setup has two steps. First find where the curves cross by setting f(x) = g(x) and solving for x — those x values are your limits of integration. Second, figure out which function is on top in between those limits, usually by testing a point, and integrate (top minus bottom) dx across that interval. The subtraction matters because it cancels out whatever area lies below both curves, leaving only the sliver of space that belongs strictly between them.

This is the direct extension of area under a curve from earlier integration work, just with a moving baseline instead of the x-axis. It sets up everything that follows in this unit: volumes of revolution and washers reuse this same top-minus-bottom sliver, just rotated into a disk or ring instead of left flat. Getting comfortable finding intersection points and checking which curve is on top here pays off immediately once solids of revolution show up.