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Area Between Curves
Multiple Intersections

Master the technique of finding areas between curves that cross each other more than twice, splitting into subregions and integrating each piece.

AP Calculus AB

What Does "Area Between Curves" Mean?

Given two continuous functions \( f(x) \) and \( g(x) \) on an interval \([a, b]\), the area between them measures the total region enclosed between the two graphs, regardless of which function is on top.

But what happens when the curves switch positions: sometimes \( f \) is above \( g \), sometimes below? That is exactly the problem we solve in this lesson.

Visualizing Multiple Intersections

Why Simple Subtraction Fails

If you blindly compute \( \displaystyle\int_a^c [f(x) - g(x)]\,dx \) across all three intersection points, positive and negative pieces cancel out, giving you a number smaller than the true area.

✕ Wrong Approach
One integral from first to last intersection: areas cancel.
✓ Correct Approach
Split at every intersection, use absolute value or swap order.

The Core Formula

On each subinterval \([x_k, x_{k+1}]\), one function stays on top. So the absolute value becomes a simple subtraction: top minus bottom.

Splitting Into Subregions

Two Intersections vs. Three or More

Two Intersections Three+ Intersections
Regions 1 enclosed region 2 or more enclosed regions
Top/Bottom Same function on top throughout Functions swap roles at each crossing
Formula \(\displaystyle\int_a^b [f(x)-g(x)]\,dx\) Sum of integrals per subinterval
Test Point One test point suffices Test point needed per subinterval

How to Apply the Method

1
Set \( f(x) = g(x) \) and solve for all intersection points \( x_1, x_2, \ldots, x_n \).
2
Order the intersections from left to right: \( x_1 < x_2 < \cdots < x_n \).
3
Test a sample point in each subinterval to determine which function is greater.
4
Set up the integral for each subinterval: \( [\text{top}] - [\text{bottom}] \).
5
Evaluate and sum all subintegrals. The total is the area.

Example 1: \( y = x^3 - x \) and \( y = 0 \)

Find the total area enclosed between \( y = x^3 - x \) and the \( x \)-axis.

Example 1: Solution

Example 2: \( y = \sin x \) and \( y = \cos x \)

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

Example 2: Evaluating Each Piece

Example 3: \( y = x^3 \) and \( y = x \)

Find the total enclosed area between \( y = x^3 \) and \( y = x \).

The Absolute Value Shortcut

Instead of manually determining which function is on top in each subinterval, you can write:

By Hand
Split into subintervals, determine top/bottom, integrate each piece.
With Calculator
Enter \( \int_a^b |f(x)-g(x)|\,dx \) directly. Fast and reliable.

Common Mistakes

✕ Forgetting to Split
Integrating \( f(x) - g(x) \) across all intersections without splitting produces cancellation and a wrong (smaller) answer.
✕ Missing an Intersection
Failing to find all solutions of \( f(x) = g(x) \) means you miss a subregion entirely.
✕ Wrong Top/Bottom
Subtracting in the wrong order gives a negative integral. Always test a sample point per subinterval.

Summary of the Method

Step Action Purpose
1 Solve \( f(x)=g(x) \) Find all intersection points
2 Order \( x_1 < x_2 < \cdots \) Define subintervals
3 Test a point per interval Determine top vs. bottom
4 Integrate \( [\text{top}] - [\text{bottom}] \) Get area of each region
5 Sum all pieces Total enclosed area

Question 1 of 5

The curves \( y = x^2 \) and \( y = x \) intersect at \( x = 0 \) and \( x = 1 \). What is the area between them?

Question 2 of 5

When finding the area between curves that cross at three points, how many separate integrals do you need?

Question 3 of 5

What is the total area between \( y = x^3 - x \) and \( y = 0 \)?

Question 4 of 5

Why do we use a test point in each subinterval when computing area between curves?

Question 5 of 5

The area between \( y = \sin x \) and \( y = \cos x \) from \( 0 \) to \( 2\pi \) equals: