Area between curves is one of 7 applications formulas in the calculus section of this library, and it is used at ap · university level.
Why area between curves works
At each input the vertical gap between the two graphs is the height of a very thin rectangle, and its width is the tiny step in x. Adding every one of those rectangles is what the integral does, so only the distance between the curves matters and not how high either one sits.
What each symbol means
$a,b$ bound the region and $f,g$ are the curves.
Area between curves: when it holds
Split the integral where the top curve changes; absolute value ensures geometric area is nonnegative.
When it stops applying
Vertical strips assume each boundary is a single function of x. For a sideways parabola a vertical strip meets the curve twice, so the setup miscounts; integrating in the other variable with left and right boundaries fixes it.
Area between curves: a worked example
Between $y=x$ and $y=x^2$ on $[0,1]$, $A=\int_0^1(x-x^2)dx=1/6$.
The mistake to avoid
What people do: Keeping one fixed subtraction order across an interval where the two curves cross.
Why it goes wrong: Past the crossing the upper curve becomes the lower one, so that stretch is counted as negative area and the total comes out too small.
Do this instead: Solve for the crossing points, split the integral there, and subtract lower from upper separately on each piece.
Area between curves: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $a,b$ bound the region and $f,g$ are the curves.
- Check the conditions before substituting. Split the integral where the top curve changes; absolute value ensures geometric area is nonnegative.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most calculus slips.
Where this formula fits
- Subject
- Calculus formulas — 42 entries in this library
- Topic
- Applications
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where area between curves comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Fundamental Theorem of Calculus — the lesson behind this formula: connect accumulated area with instantaneous change.
- Calculus Calculator — check your substitution and the value it produces.
- Study calculus — the subject guide that explains the ideas these formulas compress.
- Calculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 42 calculus formulas — the full grouped reference, or the complete formula library.
Questions about area between curves
Do I need the absolute value if I already know which curve is on top?
No. Just subtract the lower from the upper on that interval. The absolute value is only insurance in case the order flips somewhere you did not notice.
What if the whole region sits below the horizontal axis?
It changes nothing. The integrand measures the gap between the two curves, so a region entirely under the axis still produces a positive area.
How do I find the limits when the problem gives none?
Set the two expressions equal and solve. The inputs where the curves meet are the natural left and right edges of the enclosed region.
Why is the region between y = x and y = x squared only one sixth?
Because the gap is never wider than 0.25, which it reaches halfway along, and it closes to nothing at both ends, so the sliver really is that thin.