8.6 Finding the Area Between Curves That Intersect at More Than Two Points

Find all intersection points first

When two curves cross more than twice, set f(x) = g(x) and solve for ALL x-values where they intersect. These points split the interval into subintervals where one function is consistently on top. Sketching a graph helps you see which curve is greater on each piece — never assume the same function stays on top across the whole interval.

Sum the absolute differences

On each subinterval, the area is ∫|top − bottom| dx. Since the 'top' function may switch, you must write a separate integral for each subinterval using the correct order of subtraction. Total area = sum of these integrals: ∫(f−g)dx + ∫(g−f)dx + ... Adding them (not subtracting) gives the true enclosed area, even where one piece would otherwise be negative.

Watch for symmetry and multiple regions

Curves may create several separate enclosed regions between consecutive intersection points. Each region's area must be computed with non-negative integrands. A common error is integrating (f−g) across the entire domain without adjusting signs — this can cause regions to cancel out instead of adding, giving an incorrect (too small) total area.

Key terms

First step when curves intersect more than twice

Solve f(x) = g(x) to find ALL intersection points, splitting the domain into subintervals.

Area formula on each subinterval

∫ (top function − bottom function) dx, using whichever function is greater on that piece.

Total area with multiple intersections

Sum of the separate integrals from each subinterval, each using correct top/bottom order.

Common mistake to avoid

Integrating (f−g) across the whole domain without splitting can cause cancellation and an incorrect smaller area.

Why sketch the graph?

A sketch shows which function is on top in each region, helping set up correct integrands and bounds.

Try an AP-style question

What is the total area between y = x³ and y = x?

1. 1/2
2. 0
3. 1
4. 1/4