Area Between Curves Formula

Calculus 2 · Axiom Academy

LESSON Area Between Curves Formula Find the area trapped between two functions by integrating the vertical gap — top minus bottom — across the interval. 1. The Big Idea: Stack Up Strips Identify the upper function f(x) and the lower function g(x) . Slice the region into thin vertical strips. A strip at position x has width dx and height f(x)-g(x) , so its area is . Sweeping a strip across [a,b] and adding the slivers builds the whole region — that running sum is the integral. The single most important decision is which function is the upper one. Don't go by which formula "looks bigger" — go by which graph sits higher on the interval. The animation drops a measuring stick at several values of x : each stick spans the vertical distance from the lower curve up to the upper curve. That distance is exactly the integrand f(x)-g(x) . The graph that sits higher on the interval. Its y -value is the top of every strip. The graph that sits lower. Its y -value is the bottom of every strip. Top minus bottom, f(x)-g(x) , measured straight up at each x . This is the strip height. Plug a test x into both functions; the larger output is the upper one there. At an intersection the curves swap roles — what was on top drops underneath. If you integrate one fixed difference straight through a crossing, the two pieces fight each other and partially cancel. The fix: find where they cross by solving f(x)=g(x) , then split into separate integrals and use the correct top minus bottom on each side.

This is the written version of the interactive lesson above. See the full Calculus 2 course.