How to find the area between two curves?

The basic principle: upper curve minus lower curve

Determining the area between two curves involves finding the vertical distance between them at every point across an interval. If f(x) sits above g(x) on an interval from a to b, the height of a tiny vertical strip of that region is f(x) minus g(x).

Integrating this height from a to b sums all those infinitesimally thin strips. The formula is the definite integral from a to b of (f(x) - g(x)) dx. Because area represents a physical space, the result must always be positive.

Step-by-step method

First, find where the two curves intersect by setting f(x) equal to g(x) and solving for x. These solutions provide your integration limits, a and b. If a problem specifies vertical boundary lines like x = 0 or x = 3, check whether the curves cross each other between those lines as well.

Second, determine which curve is on top on that interval. You can sketch the curves or test a single x-value between your limits. If f(test) is larger than g(test), then f(x) is the upper curve.

Third, set up the integral as the upper function minus the lower function. Remember to distribute any negative sign across every term of the lower function before finding the antiderivative and evaluating at the endpoints.

Handling crossing curves and boundary lines

Sometimes two curves intersect more than twice and switch positions. For example, y = x^3 and y = x enclose two separate regions: from x = -1 to 0, and from x = 0 to 1. On the left interval, x^3 is above x; on the right interval, x is above x^3.

Whenever curves cross, split the problem into separate integrals for each section. Calculate each integral with its respective upper minus lower function, then add the positive areas together. For regions enclosed by a curve, a tangent line, and an axis, sketch the region carefully to identify the boundary points before integrating.