Area between curves expressed as functions of x — AP Calculus AB
1. Core Definition of Area Between Curves ★★☆☆☆ ⏱ 3 min
The area between two curves $y=f(x)$ and $y=g(x)$ bounded between vertical lines $x=a$ and $x=b$ is the total positive area of the enclosed region. This topic makes up 7-10% of the total AP Calculus AB exam score, appearing in both multiple-choice and free-response sections. Unlike net area under a single curve (where area below the x-axis counts as negative), area between two curves is always positive by definition. We approximate area with vertical slices of width $\\Delta x$, then take the limit of the Riemann sum to get a definite integral.
2. Basic Area Formula for Fixed Intervals ★★☆☆☆ ⏱ 4 min
When one function is always above the other on the entire interval $[a,b]$, the height of each vertical slice is the difference between the y-value of the upper function and the y-value of the lower function. Summing these areas gives the Riemann sum, which becomes the definite integral in the limit:
A = \int_a^b \left[ f_{\text{upper}}(x) - g_{\text{lower}}(x) \right] dx
The x-axis itself is a curve $y=0$, so the area between a curve and the x-axis is just a special case of this general formula.
Exam tip: If an AP question says "set up but do not evaluate the integral," stop after writing the integral expression to avoid wasting time on extra work.
3. Finding Bounds from Intersection Points ★★★☆☆ ⏱ 3 min
Many AP problems do not give explicit interval bounds $[a,b]$, and instead ask for the area of the region enclosed by two curves. In these cases, the bounds are the x-coordinates of the intersection points of the two curves. After solving $f(x) = g(x)$, sort the solutions to get the lower bound $a$ (smallest x) and upper bound $b$ (largest x). Always confirm which function is upper on the interval between intersections, do not assume based on leading coefficients or other general properties.
Exam tip: Always check for extraneous solutions when solving for intersections, especially with square roots or rational functions. Confirm each solution in both original functions before using it as a bound.
4. Regions with Changing Upper/Lower Boundaries ★★★★☆ ⏱ 4 min
When two curves intersect more than once within the interval of interest, the order of the upper and lower functions switches between consecutive intersection points. We cannot use a single integral over the entire interval, because negative area from one subinterval will cancel positive area from another, giving an incorrect final result. Instead, sort all intersection points by x, split the original interval into subintervals where the upper/lower order is constant, calculate area for each subinterval, then add the results.
Exam tip: When adding areas of multiple subintervals, always add the results. Do not subtract them, because each integral already gives positive area for its subinterval.
Common Pitfalls
Why: Students mix up bound order and forget area must always be positive.
Why: Students forget that switching upper/lower changes the sign of the integrand, leading to incorrect cancellation of positive and negative areas.
Why: Students rush to set up the integral without checking which function is upper on the interval.
Why: Students confuse problems asking for area of a fully enclosed region with problems asking for area between two given vertical lines.
Why: Students forget that the x-axis is the curve $y=0$, so it acts as the upper function when $f(x)$ is negative.