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

AP® Calculus AB & BC | Advanced Area Problems

Why This Matters: Real curves don't always intersect just twice! When two curves cross at multiple points, they create MULTIPLE regions, and each region needs separate treatment. This is where area problems get challenging—and where AP® exams love to test your understanding! You must find ALL intersection points, determine which function is on top in EACH region, and sum the areas. This skill combines algebra, analysis, and integration. Master this and you've conquered one of the toughest area topics!

🎯 The Challenge: Multiple Regions

THE PROBLEM

When curves intersect at more than two points, they create multiple enclosed regions. Each region may have a DIFFERENT function on top!

Key Insight:

You CANNOT use a single integral! You must:

Find ALL intersection points
Identify each separate region
Determine which function is on top in each region
Set up a separate integral for each region
Add all the areas together

📐 The General Formula

Total Area with Multiple Intersections

THE APPROACH:

If curves intersect at points \(x = a, b, c, d, \ldots\) (ordered left to right):

\[ A_{\text{total}} = A_1 + A_2 + A_3 + \cdots \]

Where each \(A_i\) is the area of one region

For each region:

\[ A_i = \int_{x_i}^{x_{i+1}} |f(x) - g(x)| \, dx \]

Or more practically:

\[ A_i = \int_{x_i}^{x_{i+1}} [\text{top} - \text{bottom}]_i \, dx \]

📝 Critical Point: The "top" and "bottom" functions can SWITCH from region to region! Always check which is which in each interval.

📋 Complete Step-by-Step Process

The Systematic Method

The 7-Step Approach:

Find ALL intersection points: Solve \(f(x) = g(x)\) completely
- Don't stop at first solution!
- Check the entire relevant domain
Order the intersection points: List as \(x_1 < x_2 < x_3 < \cdots\)
Identify each region: Between consecutive intersection points
For EACH region, determine top/bottom: Test a point in that interval
Set up integral for each region: \(\int_{x_i}^{x_{i+1}} [\text{top} - \text{bottom}]\,dx\)
Evaluate each integral: Find each area separately
Add all areas: \(A_{\text{total}} = A_1 + A_2 + A_3 + \cdots\)

📖 Comprehensive Worked Examples

Example 1: Three Intersection Points

Problem: Find the area between \(f(x) = x^3 - 4x\) and \(g(x) = 0\) (the x-axis).

Solution:

Step 1: Find ALL intersection points

\[ x^3 - 4x = 0 \]

\[ x(x^2 - 4) = 0 \]

\[ x(x-2)(x+2) = 0 \]

Intersection points: \(x = -2, 0, 2\)

Step 2: Identify regions

Region 1: \([-2, 0]\)

Region 2: \([0, 2]\)

Step 3: Determine top/bottom in each region

Region 1 \([-2, 0]\): Test \(x = -1\)

\(f(-1) = (-1)^3 - 4(-1) = -1 + 4 = 3 > 0\)

So \(f(x)\) is on top

Region 2 \([0, 2]\): Test \(x = 1\)

\(f(1) = 1 - 4 = -3 < 0\)

So x-axis (0) is on top, \(f(x)\) is below

Step 4: Set up integrals

\[ A_1 = \int_{-2}^0 (x^3 - 4x) \, dx \]

\[ A_2 = \int_0^2 [0 - (x^3 - 4x)] \, dx = \int_0^2 (4x - x^3) \, dx \]

Step 5: Evaluate

Region 1:

\[ A_1 = \left[\frac{x^4}{4} - 2x^2\right]_{-2}^0 = 0 - (4 - 8) = 4 \]

Region 2:

\[ A_2 = \left[2x^2 - \frac{x^4}{4}\right]_0^2 = (8 - 4) - 0 = 4 \]

Step 6: Add areas

\[ A_{\text{total}} = A_1 + A_2 = 4 + 4 = 8 \]

ANSWER: Total area = 8 square units

Example 2: Sine and Cosine (Multiple Crossings)

Problem: Find the area between \(y = \sin x\) and \(y = \cos x\) from \(x = 0\) to \(x = 2\pi\).

Step 1: Find intersection points in \([0, 2\pi]\)

\[ \sin x = \cos x \]

\[ \tan x = 1 \]

Solutions: \(x = \frac{\pi}{4}, \frac{5\pi}{4}\)

Step 2: Identify regions

Region 1: \([0, \frac{\pi}{4}]\)

Region 2: \([\frac{\pi}{4}, \frac{5\pi}{4}]\)

Region 3: \([\frac{5\pi}{4}, 2\pi]\)

Step 3: Determine top/bottom

Test \(x = 0\): \(\cos 0 = 1 > \sin 0 = 0\) → cos on top

Test \(x = \pi\): \(\sin \pi = 0 > \cos \pi = -1\) → sin on top

Test \(x = \frac{3\pi}{2}\): \(\cos \frac{3\pi}{2} = 0 > \sin \frac{3\pi}{2} = -1\) → cos on top

Step 4: Set up and evaluate

\[ A = \int_0^{\pi/4} (\cos x - \sin x)\,dx + \int_{\pi/4}^{5\pi/4} (\sin x - \cos x)\,dx + \int_{5\pi/4}^{2\pi} (\cos x - \sin x)\,dx \]

After evaluation: \(A = 4\sqrt{2}\)

Example 3: Polynomial Curves (Four Intersections)

Problem: Find area between \(f(x) = x^3 - 3x\) and \(g(x) = x\).

Find intersections:

\[ x^3 - 3x = x \]

\[ x^3 - 4x = 0 \]

\[ x(x^2 - 4) = 0 \]

Points: \(x = -2, 0, 2\)

Regions and setup:

Test shows:

\([-2, 0]\): \(g(x) > f(x)\)
\([0, 2]\): \(f(x) > g(x)\)

\[ A = \int_{-2}^0 [x - (x^3-3x)]\,dx + \int_0^2 [(x^3-3x) - x]\,dx \]

\[ = \int_{-2}^0 (4x - x^3)\,dx + \int_0^2 (x^3 - 4x)\,dx \]

By symmetry (odd function): \(A = 2\int_0^2 (x^3 - 4x)\,dx = 8\)

🔑 Special Techniques and Shortcuts

Using Absolute Value:

Alternative Approach:

If you're finding total area regardless of position:

\[ A = \int_a^b |f(x) - g(x)| \, dx \]

This automatically handles switching, but requires knowing where to split!

Using Symmetry:

Even functions: If symmetric about y-axis, calculate one side and double
Odd functions: Net signed area is zero, but absolute areas aren't!
Periodic functions: One period may be enough

🔍 Finding ALL Intersection Points

Common Methods & Pitfalls:

Algebraic Methods:

Factoring: Factor completely, find all roots
Quadratic formula: For degree 2 equations
Substitution: For trig or exponential equations
Graphing calculator: Use to find approximate intersections

⚠️ Don't miss solutions!

Check the ENTIRE domain
For trig: find all solutions in given interval
For polynomials: degree tells max number of intersections
Graph to verify you found them all

💡 Essential Tips & Strategies

✅ Success Strategies:

Sketch the curves: Visual helps identify all regions
List intersections in order: Left to right prevents confusion
Test each region: Never assume which is on top
Use calculator strategically: Find intersections, verify regions
Check your work: Area must be positive!
Look for symmetry: Can save significant work
Keep track of signs: Especially when integrating
Add carefully: Multiple integrals = multiple chances for error

🔥 Quick Checks:

Number of regions: = (number of intersections) - 1
Each integral positive: If any is negative, you switched top/bottom
Total must be positive: Area is always non-negative
Verify intersection count: Polynomial degree gives max intersections

❌ Common Mistakes to Avoid

Mistake 1: Missing intersection points (not solving completely)
Mistake 2: Using single integral when curves cross
Mistake 3: Not testing which function is on top in each region
Mistake 4: Assuming functions keep same position
Mistake 5: Arithmetic errors when adding multiple areas
Mistake 6: Wrong limits on integrals
Mistake 7: Not ordering intersection points
Mistake 8: Getting negative area (sign error in subtraction)
Mistake 9: Missing a region entirely
Mistake 10: Calculator errors (wrong mode, syntax)

📝 Practice Problems

Find the total area for these curves:

\(f(x) = x^2\) and \(g(x) = x^3\) (they intersect at 3 points)
\(y = \sin x\) and \(y = 0\) from \(x = 0\) to \(x = 2\pi\)
\(f(x) = x^3 - x\) and \(g(x) = 0\)
\(y = x^2 - 4\) and \(y = 0\) (find all enclosed regions)

Answers:

\(\frac{1}{12}\) square units (intersect at \(x = 0, 1\); also at \(x = -1\) if considering)
4 square units (two regions: \([0,\pi]\) and \([\pi, 2\pi]\))
\(\frac{1}{2}\) square units
\(\frac{32}{3}\) square units

✏️ AP® Exam Success Tips

What AP® Graders Look For:

Show finding ALL intersections: Solve equation completely
Identify all regions: List them clearly
Show testing: Which function is on top in each region
Separate integrals: One for each region
Show each evaluation: Don't skip regions
Add areas clearly: Show \(A_1 + A_2 + A_3 = \text{total}\)
Positive result: All areas must be positive
Include units: "square units"

💯 Exam Strategy:

Read carefully: How many intersections mentioned?
Solve \(f(x) = g(x)\) COMPLETELY
Order intersection points
Sketch if time permits (prevents errors)
For EACH region between consecutive points:
- Test which is on top
- Write separate integral
Evaluate each integral
Add all areas
Check: Is total positive?

⚡ Quick Reference Guide

MULTIPLE INTERSECTIONS ESSENTIALS

The Process:

Find ALL intersections: \(f(x) = g(x)\)
Order them: \(x_1 < x_2 < x_3 < \cdots\)
For each region \([x_i, x_{i+1}]\):
- Test which is on top
- Integrate: top - bottom
Add all areas

Remember:

Different regions, different integrals!
Always test which is on top
Each area must be positive
Total = sum of all regions

Master Multiple Intersection Problems! When curves intersect at MORE than two points, they create MULTIPLE regions—each requiring a separate integral. The complete process: (1) Find ALL intersection points by solving \(f(x) = g(x)\) completely—don't miss any! (2) Order intersections left to right: \(x_1 < x_2 < x_3\), etc. (3) For EACH region between consecutive intersection points, determine which function is on top by testing a point. (4) Set up separate integral for each region: \(\int_{x_i}^{x_{i+1}}[\text{top} - \text{bottom}]\,dx\). (5) Evaluate each integral (each must be positive!). (6) Add all areas: \(A_{\text{total}} = A_1 + A_2 + A_3 + \cdots\). Critical: Functions can SWITCH positions—what's on top in one region may be on bottom in the next! Common mistake: using a single integral when curves cross multiple times. Shortcuts: use symmetry when applicable, absolute value method \(\int|f-g|\,dx\). This is a major AP® exam challenge—requires careful algebra, systematic analysis, and multiple integrations. Practice until you can handle 3+ intersection problems confidently! 🎯✨