Welcome to the Area Between Curves!

In Unit 6, you learned how to find the "area under a curve" by integrating a function from point \(a\) to point \(b\). Now, we are going to level up! In Unit 8, we look at the space trapped between two different functions. Think of it like finding the area of a piece of fabric cut between two wavy lines. This concept is the foundation for many real-world applications, from designing architectural arches to calculating the total profit between cost and revenue models.

8.4 Finding the Area Between Curves (Functions of \(x\))

When we have two functions, \(f(x)\) and \(g(x)\), and we want to find the area bounded between them from \(x = a\) to \(x = b\), we use a simple principle: Top Minus Bottom.

If \(f(x) \geq g(x)\) on the entire interval \([a, b]\), the area \(A\) is given by:

\(A = \int_{a}^{b} [f(x) - g(x)] \, dx\)

Step-by-Step Process:

  1. Identify the "Top" and "Bottom" functions: If you aren't sure which is which, pick a number between \(a\) and \(b\) and plug it into both functions. The one with the higher \(y\)-value is the "Top."
  2. Find the Limits of Integration: If the problem doesn't give you \(a\) and \(b\), you must find where the curves intersect by setting \(f(x) = g(x)\) and solving for \(x\).
  3. Set up the Integral: Place the Top function first, subtract the Bottom function, and wrap them in parentheses.
  4. Integrate: Use the Fundamental Theorem of Calculus (or your calculator if permitted).

Quick Review: Why "Top Minus Bottom"? Because the integral of the top function gives the area all the way down to the \(x\)-axis. Subtracting the integral of the bottom function "cuts out" the empty space below, leaving only the area between them!

Key Takeaway: Always ensure your integrand is (Higher Function) \(-\) (Lower Function) to ensure your area is positive. Area in this context is never negative!

8.5 Finding the Area Between Curves (Functions of \(y\))

Sometimes, functions are given as \(x = f(y)\) and \(x = g(y)\), or the shapes are stacked side-by-side rather than top-to-bottom. In these cases, it is much easier to integrate with respect to \(y\).

For functions of \(y\), we use the principle: Right Minus Left.

If \(f(y) \geq g(y)\) on the interval \([c, d]\), the area \(A\) is:

\(A = \int_{c}^{d} [f(y) - g(y)] \, dy\)

Note: The limits of integration (\(c\) and \(d\)) are now \(y\)-values on the vertical axis!

When should you integrate with respect to \(y\)?

  • When the equations are already solved for \(x\) (e.g., \(x = y^2\)).
  • When the "top" and "bottom" functions change halfway through the region, but the "right" and "left" functions stay the same.

Common Mistake: Using \(x\)-values as limits when integrating with \(dy\). Always double-check that your limits match your variable of integration!

Key Takeaway: For \(dy\) integrals, look at the graph from the right side. The "Top" is the function furthest to the Right.

8.6 Curves Intersecting at More Than Two Points

Sometimes curves cross each other like a DNA strand. If \(f(x)\) is on top for a while, but then \(g(x)\) takes over, you cannot find the area with a single simple integral. You have to split the region into pieces.

The Absolute Value Definition: Technically, the area between any two curves is defined as:

\(Area = \int_{a}^{b} |f(x) - g(x)| \, dx\)

How to solve these:

  1. Find all intersection points: Set \(f(x) = g(x)\) to find every point where they cross.
  2. Break it up: Set up separate integrals for each sub-region.
  3. Example: If \(f\) is above \(g\) from \(x=1\) to \(x=3\), and \(g\) is above \(f\) from \(x=3\) to \(x=5\):
    \(Area = \int_{1}^{3} [f(x) - g(x)] \, dx + \int_{3}^{5} [g(x) - f(x)] \, dx\)

Did you know? On the calculator-permitted section of the AP Exam, you can simply type the absolute value integral \(\int_{a}^{b} |f(x) - g(x)| \, dx\) directly into your calculator, and it will handle the "splitting" for you!

Key Takeaway: If the curves cross, you must account for the change in orientation. Total area is the sum of the absolute areas of all individual sections.

Calculator Skills & Exam Tips

In Section I Part B and Section II Part A, a graphing calculator is required. Here is how to use it for Area Between Curves:

  • Finding Zeros: Use the "Intersect" or "Solver" feature to find \(a\) and \(b\). Pro Tip: Don't round these intermediate values! Store them as variables (like \(A\) and \(B\)) in your calculator.
  • Numerical Integration: Use the fnInt or integral tool. You must write the setup (the integral notation) on your paper, even if the calculator does the work.
  • Rounding: The AP standard is to round or truncate to three decimal places at the very end.
  • Units: If the problem provides units (like "Area in square feet"), make sure to include them in your final answer.

Don't worry if this seems tricky at first! The hardest part is usually just the algebra of finding where the curves cross. Once you have your "Top," "Bottom," and "Limits," the calculus is just a definite integral like you've done before.


Summary Checklist

1. Area with \(dx\): \(\int (Top - Bottom) \, dx\)

2. Area with \(dy\): \(\int (Right - Left) \, dy\)

3. Intersecting curves: Split the integral at the intersection points or use absolute value.

4. Cross-Reference: Once you master finding areas, you'll use these same boundaries to find Volumes in the next chapters (8.7 - 8.12)!