Area Between Curves

AP Calculus · Axiom Academy

Finding the region between two functions Step 1: Setting Up the Problem When two curves intersect, they enclose a region. To find the area of this region, we integrate the difference between the functions. Step 2: Finding Intersection Points Before we can integrate, we must find where the curves meet. These intersection points become our bounds of integration. Set and solve for x to find the limits a and b. Example: If , find intersections: Step 3: Determining Which Function is on Top The upper and lower functions might switch roles. Always check which function is greater in each interval. At x = test point in [a,b], evaluate both functions. The larger value indicates the upper curve. Step 4: Computing the Integral Once you have the setup, evaluate the definite integral: Sketch both curves to visualize the region Find intersection points by solving f(x) = g(x) Determine which curve is above the other in each interval Evaluate the integral using the Fundamental Theorem of Calculus

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