8.4 Finding the Area Between Curves Expressed as Functions of x

Setting up the area integral

When two curves are given as functions of x, the area between them from x = a to x = b is A = ∫[a,b] (top − bottom) dx, where 'top' is whichever function has the greater y-value on that interval. Always identify which curve is on top before integrating — sketching a graph helps confirm this.

Finding intersection points

The bounds a and b are usually the x-coordinates where the curves intersect, found by setting f(x) = g(x) and solving. If no bounds are given explicitly, you must solve for intersections yourself to determine the limits of integration.

Curves that switch positions

If the top and bottom curves swap somewhere in the interval, split the integral at each intersection point and use absolute value or separate integrals with the correct (top − bottom) for each piece. Adding the areas of each piece gives the total enclosed area.

Key terms

Area between two curves formula (functions of x)

A = ∫[a,b] (top function − bottom function) dx, where a and b are the x-values bounding the region.

How to find bounds a and b

Set f(x) = g(x) and solve for x; these solutions are the intersection points used as integration limits.

What to do if curves cross within the interval

Split the integral at each intersection point, so the correct (top − bottom) order is used in each subinterval, then sum the areas.

Why sketch the curves first?

A sketch helps determine which function is on top (greater y-value) over each part of the interval before setting up the integral.

Area between y = f(x) and y = g(x) on [1,3] if f ≥ g there

A = ∫[1,3] (f(x) − g(x)) dx

Try an AP-style question

What is the area between y = x and y = x² on [0, 1]?

1. 1/6
2. 5/6
3. 1/3
4. 1/2