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

Area Between Curves That Intersect More Than Twice

When curves cross each other at several points, the difference $f(x) - g(x)$ changes sign. A definite integral by itself gives signed area, which means positive and negative regions can cancel out.

But when the question says find the area between the curves, it wants total area, which must be positive.

The general formula is:

$$\text{Area} = \int_a^b |f(x) - g(x)| \, dx$$

That absolute value guarantees positivity.
In practice, we usually split the integral instead of integrating the absolute value.

Why One Integral Is Not Enough

Picture two curves crossing three times. That creates multiple “lobes” of area.

$y = x^3 - 3x$ and $y = 0$ on $[-\sqrt{3}, \sqrt{3}]$

In this example, the cubic crosses the x-axis at $-\sqrt{3}$, $0$, and $\sqrt{3}$, creating three separate regions.

If you computed

$$\int_{-\sqrt{3}}^{\sqrt{3}} (x^3 - 3x)\,dx$$

you’d get 0 because the left and right areas cancel.
But the total area is definitely not zero.

That cancellation is exactly what we’re avoiding.

The Complete Strategy

This process is mechanical. Follow it every time.

1. Find All Intersection Points

Set the equations equal:

$$f(x) = g(x)$$

Solve algebraically.
These x-values break the interval into subintervals.

If bounds are given, check whether intersections occur inside them.

Missing even one intersection usually ruins the entire answer.

2. Choose Vertical or Horizontal Slices

Most problems use vertical slices.

If the region stacks vertically, use vertical slices.
If the region overlaps vertically and is easier left-to-right, use horizontal slices.

3. Determine Which Function Is On Top

Between each pair of intersection points:

• Sketch a quick graph, or
• Plug in a test value.

The top function may change from interval to interval.
That’s the whole reason we split the integral.

4. Split the Integral

If intersections occur at
$x_1 < x_2 < x_3 < x_4$,

then:

$$\text{Area} = \int_{x_1}^{x_2} (\text{top} - \text{bottom})\,dx + \int_{x_2}^{x_3} (\text{top} - \text{bottom})\,dx + \int_{x_3}^{x_4} (\text{top} - \text{bottom})\,dx$$

Each piece must use the correct order.

Then add the results.

On a no-calculator FRQ, splitting like this is the safest method and earns structure points cleanly.

Using Absolute Value Instead

You can write:

$$\text{Area} = \int_a^b |f(x) - g(x)|\,dx$$

This works perfectly on calculator sections.
But on non-calculator work, absolute values often force you to split anyway.

So conceptually, both methods are the same. Splitting just makes the sign change explicit.

When This Shows Up

You’ll see this when:

• Solving $f(x) = g(x)$ gives three or more solutions.
• A polynomial crosses another polynomial multiple times.
• A trig function oscillates and intersects a line repeatedly.
• The graph clearly shows multiple enclosed regions.

AP questions often include symmetry. If a region looks symmetric about the y-axis, compute one side and double it only if the top/bottom relationship is the same on both sides.

Common Mistakes

• Using one integral when curves switch order. This gives cancellation.
• Subtracting in the wrong order. If total area comes out negative, something flipped.
• Forgetting intersections inside given bounds.
• Assuming symmetry without checking which function is on top.

A 10-second sketch prevents most of these.