Area Between Curves

You’ll be able to

Set up the integral of (top − bottom) for the area between two curves
Find intersection points to determine the limits of integration
Integrate with respect to y when curves are functions of y

Top minus bottom

The area between two curves is the integral of the upper curve minus the lower curve over the interval where they overlap: A = ∫[a to b] (top − bottom) dx. Each thin vertical strip has height (top − bottom) and width dx; adding them up gives the area. Because you subtract, the result stays positive even when the curves dip below the x-axis — the axis position is irrelevant.

Find the limits from the intersections

The limits a and b are usually the x-coordinates where the curves intersect. Set the two functions equal, f(x) = g(x), and solve to find where the region begins and ends. If the curves cross in the middle, the "top" and "bottom" swap, and you must split the integral at the crossing so you always integrate top minus bottom.

Integrating with respect to y

When the curves are more naturally written as functions of y — or the strips are horizontal — integrate with respect to y instead: A = ∫[c to d] (right − left) dy. Horizontal strips have width (right function − left function) and height dy. Choosing dx vs. dy to avoid splitting the region into many pieces is a strategic decision that can save a lot of work.

Area between curves

A = ∫[a to b] (top − bottom) dx or A = ∫[c to d] (right − left) dy

Always the larger function minus the smaller. Limits come from the intersection points of the curves.

Worked example

Find the area enclosed between y = x and y = x².

1.Find intersections: x = x² gives x² − x = 0, so x(x − 1) = 0 and x = 0, x = 1.
2.On (0, 1) test x = 0.5: y = x gives 0.5, y = x² gives 0.25, so y = x is on top.
3.Set up the integral: A = ∫[0 to 1] (x − x²) dx.
4.Integrate: [x²/2 − x³/3] from 0 to 1 = (1/2 − 1/3) − 0 = 1/6.

Answer: The enclosed area is 1/6. The line y = x lies above the parabola y = x² throughout (0, 1), so top − bottom = x − x².

Watch out

Always integrate top minus bottom, not the reverse — swapping them makes the area come out negative. If the curves cross inside the interval, split the integral at the crossing and use the correct top curve on each piece.

Checkpoint

The area between y = f(x) (upper) and y = g(x) (lower) from x = a to x = b is:

Checkpoint

At what x-values do y = x² and y = 4 intersect, setting the limits of integration?

On the exam

Before integrating, sketch the two curves and their intersections to see which is on top and whether they cross inside the interval. That quick picture prevents both limit errors and top/bottom mix-ups that graders penalize.

Answer the 2 checkpoints as you read.