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

What Area Between Intersecting Curves Means

When you compute

$$\int_a^b (f(x) - g(x))\,dx,$$

you are finding signed area. Parts where $f(x) < g(x)$ count as negative.

But area between curves means total geometric area. So the formula is

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

That absolute value forces every vertical slice to be positive because area is distance between curves.

Think of each slice as:

• Height = vertical distance = $|f(x) - g(x)|$
• Width = $dx$

If the curves cross inside $[a,b]$, the sign of $f(x) - g(x)$ changes. That’s why you usually split the integral.

Why More Than Two Intersections Changes Things

If two curves intersect only at the endpoints, one curve stays on top the whole time. Easy.

But if they intersect three or more times, the graph might look like this:

Area between curves with three intersection points

Focus on the lower diagram where the curves intersect at $a$, $b$, and $c$. The “top” curve switches at each intersection point. If you wrote one single integral from $a$ to $c$, positive and negative regions would cancel. That’s the trap.

The Complete Strategy

Here’s the clean way to handle these on a quiz or FRQ.

1. Find all intersection points

Solve

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

These x-values divide the region into subintervals.
Don’t forget to include any interval endpoints the problem gives you.

Missing one intersection is the most common mistake I see.

2. Decide which function is on top in each interval

Pick a test value between intersection points or use a graph (calculator section makes this quick).

If $f(x) - g(x) > 0$, then $f$ is on top.
If $f(x) - g(x) < 0$, then $g$ is on top.

You must check each subinterval. The order can switch more than once.

3. Write one definite integral per subinterval

On each interval, write:

$$\int (\text{top} - \text{bottom})\,dx$$

Example structure:

$$\int_{x_1}^{x_2} (f-g)\,dx + \int_{x_2}^{x_3} (g-f)\,dx + \int_{x_3}^{x_4} (f-g)\,dx$$

Then add them.

This removes the need for absolute value because you’ve handled the sign manually.

4. Evaluate and add

• Find antiderivatives.
• Evaluate each definite integral.
• Add results.

Final answer should be positive and include units squared if units are given.

On FRQs, clearly showing the split integrals usually earns setup points even before evaluating.

Vertical vs Horizontal Slices

Most problems use vertical slices:

$$\text{Area} = \int (\text{top} - \text{bottom})\,dx$$

That works when curves are given as $y = f(x)$.

If curves are easier to write as $x = f(y)$, then use horizontal slices:

$$\text{Area} = \int (\text{right} - \text{left})\,dy$$

Same idea. The subtraction always represents distance.

If the ordering switches with respect to $y$, you split in $y$ instead.

When You Must Split

You must break the integral into pieces when:

• The curves intersect inside the interval.
• The graphs visibly cross.
• $f(x) - g(x)$ changes sign.

If you ever get zero area for a region that clearly exists, that means cancellation happened and you forgot to split.

Common Mistakes That Cost Points

• Forgetting an intersection point.
• Subtracting in the wrong order on one subinterval.
• Leaving the answer negative.
• Writing one big integral when the graph clearly switches.
• Algebra mistakes when simplifying $f(x) - g(x)$.

On multiple choice, the College Board loves answers that reflect one of those exact mistakes.

Big Idea Connecting It All

A definite integral represents accumulation.
Area between curves represents accumulation of distance between functions.

When that distance changes direction, you handle it piece by piece.