Finding the Area Between Curves Expressed as Functions of x

Picture the region squeezed between two curves as a stack of extremely thin vertical rectangles. Each rectangle has width dx and height equal to the top curve's y-value minus the bottom curve's y-value at that x. A single rectangle's area is (top - bottom) dx, and the integral just adds up infinitely many of these slivers across the interval. That is why the formula is integral of (f(x) - g(x)) dx: it is not some special rule, it is the same slicing idea from basic area-under-a-curve work, just applied to the gap between two functions instead of the gap between one function and the x-axis.

Before you can set up that integral you need the limits, and those come from where the curves cross. Set f(x) = g(x), solve for x, and those solutions become your bounds of integration. You also need to know which curve is on top across that interval, since subtracting backwards gives a negative of the true area. A quick check: plug in a single x-value between the bounds and see which function gives the larger y.

This topic sits right next to finding area under one curve and the x-axis, which is really just the special case where the bottom curve is y = 0. It also sets up the next step in the unit, where instead of slicing vertically you may need to slice horizontally and integrate with respect to y, useful when curves are easier to describe as x = function of y or when top and bottom swap partway through the region.