Area between curves

Top minus bottom

You know how to find the area between a curve and the x-axis. What changes when the bottom edge is another curve instead of the axis?

The region below is cut into thin vertical strips. Drag the ochre strip across the region and watch its height, which is always the top curve minus the bottom curve, wherever the strip sits.

Drag the ochre strip. Its height is top minus bottom at that x.

Curves
Intersections
Top at the strip
Strip height
Area

Top minus bottom

A thin strip at x has height f(x)-g(x), where f is the top curve, and width dx. Adding the strips gives

A=\int_a^b\big[f(x)-g(x)\big]\,dx\qquad\text{where } f(x)\ge g(x) \text{ on } [a,b].

This works wherever the curves sit, even below the axis, because the height of a strip is a difference of two heights. The axis no longer plays any part.

1. Find the intersections

Set f(x)=g(x) and solve. These usually give a and b.

2. Decide which is on top

Test one point between the intersections. If the curves cross inside the interval, the top and bottom swap, so split the integral there.

3. Integrate top minus bottom

Simplify the difference first, then integrate.

For y=x and y=x^2, the curves meet at 0 and 1, and x\ge x^2 between them, so A=\int_0^1(x-x^2)\,dx=\tfrac12-\tfrac13=\tfrac16.

Solve it step by step

Find where the curves meet, decide which is on top, and integrate. The score counts steps you answer correctly on the first try.

Problems solved0
First-try steps0 of 0

    Common mistakes

    Bottom minus top

    Reversing the order gives a negative number. Test a point to see which curve is higher.

    Ignoring a crossing

    If the curves cross inside [a,b], one integral of f-g lets the two regions cancel. Split at the crossing and add the sizes.

    Subtracting the axis areas wrongly

    There is no need to find each curve's area to the axis. One integral of the difference does the whole job.