Area between two curves

The area between two graphs is the integral of the upper function minus the lower one.

Check which is on top on each interval, and use the $x$-coordinates of the intersections as the limits. If the curves swap places, split the interval.

Examples

1. 01For 0<x<3, which expression gives the area between these two curves?$\displaystyle y=3x,\quad y=x^2$AnswerHide

   Answer$\displaystyle \int_{0}^{3} (3x-x^2)\,dx$
2. 02The area enclosed by $y=9$ and $y=x^2$. What goes in □?$\displaystyle \int_{-3}^{3} (\square)\,dx$AnswerHide

   Answer$\displaystyle 9-x^2$
3. 03Which expression gives the area between these two curves?$\displaystyle y=4,\quad y=x^2$AnswerHide

   Answer$\displaystyle \int_{-2}^{2} (4-x^2)\,dx$
4. 04The area enclosed by $y=1$ and $y=x^2$. What goes in □?$\displaystyle \int_{-1}^{1} (\square)\,dx$AnswerHide

   Answer$\displaystyle 1-x^2$
5. 05The curves meet at x=0 and x=4, and y=4x is on top between them. What is the enclosed area?$\displaystyle y=4x,\quad y=x^2$AnswerHide

   Answer$\displaystyle \frac{32}{3}$
6. 06Which expression gives the area between these two curves?$\displaystyle y=16,\quad y=x^2$AnswerHide

   Answer$\displaystyle \int_{-4}^{4} (16-x^2)\,dx$

FAQ

Q1Which function do I subtract?Show answerHide

A

Subtract the lower function from the upper one. If they swap places inside the interval, split at the intersection and integrate each part separately.

Practice this pattern in the app, 5 problems a day.

Published 2026-09-03 · Updated 2026-09-05