Area Is a Difference
Between two curves, area is just the integral of the top curve minus the bottom curve.
The Setup
If f sits above g on an interval, a thin vertical strip has height f minus g and width dx. Adding those strips with an integral gives the total area between the curves. The formula works because negative gaps and positive gaps are never mixed, as long as the top curve really is on top.
A = \int_a^b \bigl(f(x) - g(x)\bigr)\,dxFind the Crossings
The intersection points set the limits of integration, so solve f equals g first. If the curves cross inside the interval, the region splits into pieces and each piece needs its own integral, with the roles of top and bottom swapped. Sketching first prevents the classic error of integrating the wrong difference.
f(x) = g(x)A Worked Region
Between y = x and y = x squared, the curves meet at x = 0 and x = 1, and the line is on top. The area is the integral of x minus x squared, which evaluates to one half minus one third, or one sixth. A parabola and its chord always enclose a third of the bounding rectangle, a pattern worth noticing.
\int_0^1 (x - x^2)\,dx = \frac{1}{6}Flashcard
Area between curves, f on top
Check your understanding
For the region enclosed by y = x^2 and y = x^3 between 0 and 1, which integrand gives the correct area?