Welcome to Unit 8: The Area Between Curves!

In Unit 6, you learned how to find the area between a curve and the \(x\)-axis. Now, we are going to take it a step further! In this chapter, we will learn how to find the area trapped between two different functions. Think of it like finding the area of a sandwich where the two functions are the pieces of bread.

This is a major part of Unit 8: Applications of Integration. Mastering this will help you understand how integration measures the "accumulation of difference" between two changing quantities. Don't worry if it seems a bit abstract at first—once you see the "Top minus Bottom" pattern, it becomes much easier!

Quick Review: Before we start, remember that a definite integral \(\int_{a}^{b} f(x) \, dx\) represents the accumulated area under a curve. In this chapter, we are just subtracting one area from another to find what's left in between.

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

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

If \(f(x)\) is the "upper" function (the one with higher \(y\)-values) and \(g(x)\) is the "lower" function on the interval \([a, b]\), the area \(A\) is:

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

Why does this work? Imagine the area under the top curve and the area under the bottom curve. If you subtract the bottom area from the top area, you are left with the space in between them!

Step-by-Step Process:

  1. Identify the "Top" and "Bottom": Sketch the graphs or plug in a test value between \(a\) and \(b\) to see which function is higher.
  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, then subtract the "Bottom" function.
  4. Integrate: Use the Fundamental Theorem of Calculus (or your calculator if permitted) to solve.

Common Mistake to Avoid: Never just guess which function is the "top." If you subtract them in the wrong order, you will get a negative area. Area must always be positive!

Key Takeaway: For functions of \(x\), the area is the integral of (Upper Function) - (Lower Function) with respect to \(x\).

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

Sometimes, functions are written as \(x = f(y)\) instead of \(y = f(x)\). Or, perhaps the curves are "stacked" side-to-side rather than one on top of the other. In these cases, it is much easier to integrate with respect to \(y\).

The rule here is: Right Minus Left.

If \(f(y)\) is the "right-most" function (larger \(x\)-values) and \(g(y)\) is the "left-most" function on the interval from \(y = c\) to \(y = d\), the area \(A\) is:

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

Memory Aid: Think of the number line. Higher numbers are to the Right and Top. So, we always do Higher Value minus Lower Value.

When should I use \(dy\)?

  • When the boundaries are given as \(y = c\) and \(y = d\).
  • When the equations are already solved for \(x\) (e.g., \(x = y^2\)).
  • When a horizontal slice is easier to calculate than a vertical slice.

Key Takeaway: For functions of \(y\), the area is the integral of (Right Function) - (Left Function) with respect to \(y\). Make sure your limits of integration (\(c\) and \(d\)) are \(y\)-values!

8.6 Area Between Curves That Intersect at More Than Two Points

What happens if the curves cross each other? For example, \(f(x)\) might be on top for a while, but then they intersect, and \(g(x)\) becomes the top function.

If you just integrate \(\int_{a}^{b} [f(x) - g(x)] \, dx\) over the whole interval, the "negative" area where \(g(x)\) is higher will cancel out the "positive" area. This will give you the net area, but we want the total area.

How to Solve "Crossing" Curves:

  1. Find ALL Intersection Points: Set \(f(x) = g(x)\) and find all solutions. Let's say they intersect at \(x = a, x = b,\) and \(x = c\).
  2. Split the Integral: Set up a separate integral for each region.
    Region 1: \(\int_{a}^{b} [f(x) - g(x)] \, dx\) (if \(f\) is on top)
    Region 2: \(\int_{b}^{c} [g(x) - f(x)] \, dx\) (if \(g\) is on top)
  3. Add the Results: The total area is the sum of the absolute values of the areas of each region.

Did you know? You can write this conceptually using absolute value: Area = \(\int_{a}^{c} |f(x) - g(x)| \, dx\). If you are using a graphing calculator on Section I Part B or Section II Part A, you can actually type the absolute value into the integral to get the answer in one step!

Key Takeaway: If curves cross, you must split the integral at every intersection point to ensure you are always subtracting "Lower" from "Upper."

Exam Tips and Calculator Usage

The AP Calculus AB exam has specific rules about how you present your work, especially in Unit 8.

1. The "Setup" is Everything

On the Free-Response Section (FRQ), even if you use a calculator to find the final number, you must write the integral "setup" first.
Example: Write "Area = \(\int_{0}^{2} (e^x - x^2) \, dx\)" before writing the decimal answer.

2. Calculator Rules

You are required to use your calculator to:

  • Find intersection points (zeros) of functions.
  • Calculate the numerical value of a definite integral.
Do not waste time integrating complex functions by hand on the calculator-active sections!

3. Rounding

Always round your final answer to three decimal places (standard AP convention) unless the problem tells you otherwise. Don't round your intersection points too early—keep several decimals for your limits of integration to ensure your final area is accurate.

4. Units of Measure

If the problem involves a real-world context (like square meters or gallons), make sure you include the units in your final answer. Area is always measured in square units.

Quick Review Box:
• Vertical slices: Area = \(\int (Top - Bottom) \, dx\)
• Horizontal slices: Area = \(\int (Right - Left) \, dy\)
• Intersections: Set functions equal to find your limits (\(a\) and \(b\)).
• Crossing: Split the integral or use absolute value.

Next Chapter Preview: Now that you can find the area of a 2D "sandwich," you'll soon learn how to turn that area into a 3D object in "Volumes with Cross Sections"! Keep up the great work!