8.4 Finding the Area Between Curves Expressed as Functions of x

To find the area between two curves expressed as functions of $x$, integrate the top function minus the bottom function over the interval where they bound the region: $A=\int_a^b [f(x)-g(x)]\,dx$. The limits $a$ and $b$ usually come from the x-values where the curves intersect, which you find by solving $f(x)=g(x)$. For AP Calculus, check which function is on top before writing the integrand.

Why This Matters for the AP Calculus Exam

Area between curves is one of the most common ways AP Calculus tests definite integrals as accumulation. On both the AB and BC exams, you may see these problems in multiple-choice questions and in free-response questions, and they often appear in a calculator-active context where you find intersection points numerically and then evaluate the integral. Showing a correct integral setup with proper notation, like $\int_a^b [f(x)-g(x)]\,dx$, is important for clear exam work because graders look for the structure of your solution, not just the final number.

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Key Takeaways

• Area between curves uses the integral of (top function minus bottom function): $A=\int_a^b [f(x)-g(x)]\,dx$.
• Find the limits $a$ and $b$ by setting $f(x)=g(x)$ and solving for the intersection x-values.
• Always check which curve is on top over the interval before subtracting; the top one is the larger y-value.
• If the curves swap which one is on top, split the integral or use $\int_a^b |f(x)-g(x)|\,dx$.
• On calculator-active questions, use the graph to find intersection points numerically, then evaluate the integral.
• Write a clean integral expression first, then evaluate; the setup is what shows your reasoning.

The Area Between Curves

When you have two functions and want the area of the region they enclose, you integrate the difference between them. Subtract the bottom function $g(x)$ from the top function $f(x)$, then integrate over the interval:

The "top" function is the one with larger y-values on that interval, and the "bottom" function has smaller y-values. The vertical distance between them, $f(x)-g(x)$, is the height of a thin vertical strip. Integrating adds up all those strips across the region.

To get the limits $a$ and $b$, find where the two graphs intersect by solving $f(x)=g(x)$. You can do this by hand for simple equations, but many exam problems are calculator-active, so you will often find the intersection x-values with a graphing calculator.

Worked Example: Setting Up and Evaluating

Find the area between $f(x)=e^x$ and $g(x)=x$ on the interval $[0,2]$.

First, decide which function is on top. On $[0,2]$, $e^x$ is always greater than $x$, so $f(x)$ is the top function and you subtract $g(x)$ from it.

Substitute the actual functions:

Integrate and evaluate the definite integral:

That value is the exact area of the region.

How to Use This on the AP Calculus Exam

Free Response

Area problems show up in free-response questions where you are given two functions and a graph. A common version gives one known intersection point and asks you to find the other before integrating.

Example based on a released AP Calculus AB free-response question: let $f(x)=\ln(x+3)$ and $g(x)=x^4+2x^3$. The graphs intersect at $x=-2$ and at $x=B$, where $B>0$. To find the enclosed area, first find $B$.

Since this is calculator-active, enter both functions and use the intersection feature to find $B\approx 0.781975$. Then set up and evaluate:

Write the full integral expression with correct limits and integrand before reporting the number. The setup shows your reasoning even if your final arithmetic slips.

MCQ

Multiple-choice questions may give you the region or the picture and ask for the correct integral setup rather than the final value. Focus on identifying the top function, the bottom function, and the correct limits. Watch for answer choices that flip the subtraction order or use the wrong bounds.

Common Trap

Set up the integrand as top minus bottom, not just $f(x)-g(x)$ by default. If you do not check which curve is higher, you can get a negative answer, which is a sign you subtracted in the wrong order.

Common Misconceptions

• Area is not always $\int [f(x)-g(x)]\,dx$ with the functions in the order they were named. It is top minus bottom, so you must check which one is larger on the interval.
• A negative result does not mean negative area. It means you subtracted bottom from top in reverse. Area between curves is always nonnegative.
• The limits of integration are the intersection x-values, not random endpoints. Solve $f(x)=g(x)$ unless the problem gives you fixed vertical boundaries like $x=a$ and $x=b$.
• If the curves cross inside the interval, you cannot use a single top-minus-bottom expression for the whole region. Split the integral at the crossing point or integrate the absolute value of the difference.
• Finding intersection points by tracing is not always exact. Use your calculator's intersection function for accuracy on calculator-active problems, and keep enough decimal places.

Related AP Calculus Guides

• Unit 8 Overview: Applications of Integration
• 8.1 Finding the Average Value of a Function on an Interval
• 8.7 Volumes with Cross Sections: Squares and Rectangles
• 8.2 Connecting Position, Velocity, and Acceleration of Functions Using Integrals
• 8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts
• 8.6 Finding the Area Between Curves That Intersect at More Than Two Points