Area between curves
Also known as: area between two curves
The area between two curves is found by integrating the difference between the upper and lower functions over the interval where they bound a region. For curves y = f(x) and y = g(x) with f above g, the area is the integral of f(x) − g(x) from a to b.
Finding the area between curves extends the idea of a definite integral. A single integral of f(x) gives the signed area between the curve and the x-axis; subtracting a second function replaces the axis with another boundary. The formula is A = ∫ from a to b of [f(x) − g(x)] dx, where f is the upper curve and g is the lower curve on the interval [a, b].
The procedure has three steps. First, find the limits of integration — often the intersection points, obtained by setting f(x) = g(x) and solving. Second, determine which function is on top; sketching the region or testing a sample point between the intersections settles it. Third, integrate the difference. For example, y = x and y = x² intersect at x = 0 and x = 1, and x is above x² in between, so the area is ∫ from 0 to 1 of (x − x²) dx = 1/2 − 1/3 = 1/6.
Two complications come up regularly. If the curves cross inside the interval, the region must be split at the crossing point and each piece integrated with its own upper function, because integrating a difference that changes sign produces cancellation rather than area. And when a region is bounded left and right rather than above and below, it is often easier to integrate with respect to y: A = ∫ from c to d of [right(y) − left(y)] dy. Choosing the variable that avoids splitting the region saves time.
On the AP Calculus AB exam, area between curves is a standard free-response topic and appears in the applications of integration unit. Questions frequently pair it with volumes of solids formed by revolving or cross-sectioning the same region, and calculator-active parts often ask you to find intersection points numerically before integrating.
Key takeaways
- The area between curves is ∫ from a to b of [f(x) − g(x)] dx, with f the upper function.
- Find the limits of integration by setting the two functions equal and solving for the intersection points.
- If the curves cross within the interval, split the integral at the crossing and use the correct upper function on each piece.
- For regions bounded left and right, integrate with respect to y using [right(y) − left(y)].
- AP Calculus AB commonly pairs this topic with volume problems built on the same region.