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Area Between Curves
Multiple Intersections

Master the technique of computing enclosed areas when two curves cross at more than two points, requiring you to split the integral into subregions.

AP Calculus BC

What Does "Area Between Curves" Mean?

Given two functions \( f(x) \) and \( g(x) \) on an interval \([a, b]\), the area between the curves is the total region enclosed between their graphs. When one function is always above the other, a single integral suffices:

The absolute value is critical: it ensures we always get a positive area, regardless of which curve is on top.

Why Multiple Intersections Matter

When curves intersect at more than two points, the function that is "on top" switches at each intersection. A single integral from the first crossing to the last would cancel positive and negative regions, giving the wrong answer.

Visualizing Subregions

Region 1
Between the first and second intersection: \( f(x) \geq g(x) \), so integrate \( f - g \).
Region 2
Between the second and third intersection: \( g(x) \geq f(x) \), so integrate \( g - f \).
Total Area
Add all subregion areas. Each is computed with the correct "top minus bottom" order.

The Core Formula

If \( f(x) \) and \( g(x) \) intersect at points \( x_1 < x_2 < \cdots < x_n \), the total enclosed area is:

On each subinterval \([x_k, x_{k+1}]\), one function dominates. We determine which is on top (by testing a sample point), then remove the absolute value:

Splitting at Intersection Points

Two Intersections vs. Multiple Intersections

Two Intersections Multiple Intersections
Subregions 1 region \( n - 1 \) regions
Top/Bottom Same function on top throughout Alternates at each crossing
Formula \(\int_a^b (f - g)\,dx\) \(\sum \int_{x_k}^{x_{k+1}} |f - g|\,dx\)
Risk Low: sign doesn't change High: cancellation if not split

How to Apply the Method

1
Find all intersection points. Set \( f(x) = g(x) \) and solve for \( x \). Factor or use technology if needed.
2
Order the intersections. Sort the solutions: \( x_1 < x_2 < \cdots < x_n \).
3
Test each subinterval. Pick a sample point \( c_k \in (x_k, x_{k+1}) \) and check whether \( f(c_k) > g(c_k) \) or vice versa.
4
Set up each integral. Write \(\int_{x_k}^{x_{k+1}} (\text{top} - \text{bottom})\,dx\) for each subinterval.
5
Evaluate and sum. Compute each integral separately, then add all the results.

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

Example 1: Computing the Area

Example 2: \( y = \sin x \) and \( y = 0 \) on \([0, 2\pi]\)

Example 2: Evaluating Each Region

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

Example 3: Computing the Total Area

Common Mistakes

✕ Single Integral
Using one integral from the first to last intersection without splitting: positive and negative areas cancel, yielding 0 or the wrong value.
✕ Missing Intersections
Forgetting to find all solutions of \( f(x) = g(x) \). Always factor completely or check graphically for hidden crossings.
✕ Wrong Top/Bottom
Assuming the same function is on top in every subinterval. Always test a point in each subinterval to confirm which function dominates.

Summary Checklist

✓
Solve \( f(x) = g(x) \) for all intersection points in the given domain.
✓
Split into subintervals between consecutive intersection points.
✓
Test a sample point in each subinterval to determine which function is on top.
✓
Integrate (top minus bottom) on each subinterval: always yields a positive number.
✓
Sum all subregion areas. The total is the area enclosed between the curves.

Question 1 of 5

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

Question 2 of 5

The total area between \( y = \sin x \) and \( y = 0 \) on \([0, 2\pi]\) is:

Question 3 of 5

How many subregions must you integrate when \( f(x) \) and \( g(x) \) have 4 intersection points?

Question 4 of 5

For \( y = x^3 \) and \( y = x \), the area of the region between \( x = 0 \) and \( x = 1 \) is:

Question 5 of 5

Which step is most important when finding the area between curves with multiple intersections?