A definite integral finds the area under one curve, but many real problems ask for the area trapped between two curves. The method adds only two steps to what you already know. You can check your work with the free tools on Calc Solver.
What the Area Between Curves Means
Imagine two curves drawn on the same axes, with one above the other over some interval. The area between them is the gap enclosed by the curves and the interval ends. It is found by integrating the vertical distance between the curves, not by integrating either curve alone.
Step 1: Find Where the Curves Meet
Set the two functions equal and solve to find the intersection points. These points give the interval limits. For y = x and y = x^2, solving x = x^2 gives x = 0 and x = 1, so the enclosed region runs from 0 to 1.
Step 2: Identify the Top and Bottom Curves
Test one x value inside the interval to see which curve is higher. Between 0 and 1, x is larger than x^2 (for example, at 0.5, x = 0.5 and x^2 = 0.25), so y = x is the top curve and y = x^2 is the bottom curve. Getting this backwards flips the sign of the answer.
Step 3: Integrate the Difference
Integrate (top minus bottom) over the interval: the integral from 0 to 1 of (x – x^2) dx. The antiderivative is x^2/2 – x^3/3. Evaluating from 0 to 1 gives (1/2 – 1/3) – 0 = 1/6. The area is 1/6 square units.
Common Mistakes to Avoid
First, integrating each curve separately and subtracting areas that were never bounded the same way. Second, using the wrong top curve on part of the interval: if the curves cross inside, split the integral at the crossing point. Third, dropping the absolute value idea: area is never negative, so a negative result means the top and bottom were swapped.
Practice Problems to Try
Find the area between y = x + 1 and y = x^2 from their intersections, then the area between y = sin(x) and the x-axis from 0 to pi. Sketch each region before integrating.