Area Between Curves (Integrating with Respect to x)
Quick Reference: Essential Formulas
| Formula | When to Use | Key Requirement |
|---|---|---|
| $A = \int_a^b [f(x) - g(x)]\,dx$ | Finding area between two curves | $f(x) \geq g(x)$ on $[a,b]$ (top minus bottom) |
| $A = \int_a^b \lvert f(x) - g(x) \rvert\,dx$ | When curves cross | Split at crossing points |
Memory aid: Integrate (top − bottom) from left to right.
Before You Start
Test yourself on these prerequisites. If any feel unfamiliar, follow the review link before continuing.
1. Definite Integral Evaluation: Can you compute this?
$$\int_0^2 (3x^2 - x)\,dx$$
Check Your Answer
$$\left[x^3 - \frac{x^2}{2}\right]_0^2 = \left(8 - 2\right) - 0 = 6$$
If this was difficult, review Definite Integral Evaluation first.
2. Fundamental Theorem of Calculus: Can you explain why this works?
$$\frac{d}{dx}\int_0^x f(t)\,dt = f(x)$$
Check Your Understanding
FTC Part 1 says the derivative of an accumulation function returns the original integrand. This is essential because we evaluate area integrals by finding antiderivatives.
If this concept is unclear, review FTC Part 2.
Why Subtract?
You already know that the definite integral $\int_a^b f(x)\,dx$ gives the signed area under a curve. But what if you want the area between two curves?
Picture two runners on parallel tracks. To find how far apart they are after a race, you don’t need their individual distances from the starting line. You subtract one position from the other. The same logic applies to areas: if two curves bound a region, the area between them is the difference of the areas “under” each curve.
Prerequisite Map
Legend: 🟡 Yellow = immediate prerequisites (must master) | 🟢 Green = this skill
Quick Reference
| Property | Value |
|---|---|
| Chapter | Chapter 5: Applications of Integration |
| Section | §5.1 Areas Between Curves |
| Difficulty | Beginner |
| Time | ~20 minutes |
Key Concepts
The Area Formula
If $f(x) \geq g(x)$ for all $x$ in $[a, b]$, then the area of the region bounded by $y = f(x)$ (top), $y = g(x)$ (bottom), and the vertical lines $x = a$ and $x = b$ is:
$$\boxed{A = \int_a^b [f(x) - g(x)]\,dx}$$
In words: Integrate (top minus bottom) from left to right.
Visualizing the Formula
y
| _____ f(x) (top curve)
| __/ \__
| __/ \__
| / SHADED \
| / AREA \
|/____________________|______ x
a b
_____ g(x) (bottom curve)
__/ \__
__/ \__Think of slicing the region into thin vertical rectangles of width $\Delta x$. Each rectangle has:
- Height: $f(x) - g(x)$ (top minus bottom)
- Width: $\Delta x$
- Area: $[f(x) - g(x)] \Delta x$
Summing all these rectangles and taking the limit gives the integral.
Why This Works
If both curves are above the x-axis:
$$A = \underbrace{\int_a^b f(x)\,dx}_{\text{area under } f} - \underbrace{\int_a^b g(x)\,dx}_{\text{area under } g} = \int_a^b [f(x) - g(x)]\,dx$$
The formula $\int_a^b [f(x) - g(x)]\,dx$ works even when the curves dip below the x-axis, as long as $f(x) \geq g(x)$ throughout $[a, b]$.
The Procedure
Step 1: Identify the top curve $y_T = f(x)$ and bottom curve $y_B = g(x)$.
Step 2: Determine the integration bounds $a$ and $b$ (either given or found from intersection points).
Step 3: Set up and evaluate $A = \int_a^b (y_T - y_B)\,dx$.
Common Pitfalls Table
| Mistake | What Goes Wrong | How to Fix |
|---|---|---|
| Subtracting in wrong order | Get negative area | Always check: top − bottom. Test a point to verify which is higher |
| Forgetting to find intersection points | Wrong integration bounds | If bounds not given, solve $f(x) = g(x)$ first |
| Using $\|f-g\|$ without splitting | Can’t integrate absolute value directly | Find where $f=g$, split into separate integrals |
| Assuming first function is always on top | Wrong setup | Graph the curves or test values in each subinterval |
| Forgetting parentheses when subtracting | Sign errors in integration | Write $(f(x)) - (g(x))$, not $f(x) - g(x)$ when $g$ is complex |
💡 Quick Self-Check for Sign Errors
After computing an area integral:
- Is your answer positive? Area must be positive. If negative, you subtracted wrong.
- Is it reasonable? A region between $x=0$ and $x=1$ probably has area less than $10$.
- Verify: Pick a point, compute $f(x)-g(x)$. Should be positive if $f$ is on top.
Practice Problems
For the region bounded by $y = x^2$ and $y = 4$ from $x = -2$ to $x = 2$, identify: (a) Which curve is the top curve? (b) Which curve is the bottom curve? (c) Write (but don’t evaluate) the integral for the area.
Find the area of the region bounded above by $y = 6 - x$, below by $y = 2$, and on the sides by $x = 1$ and $x = 3$.
🔄 Still confused about setting up area integrals?
- Review Definite Integral Evaluation if you’re struggling with the integration step
- Review Riemann Sums if the “stacking rectangles” picture isn’t clicking
- Try drawing the region first: sketching makes top/bottom identification easier
Find the area of the region bounded by $y = x^2 + 1$ (above) and $y = 3x - 1$ (below) from $x = 0$ to $x = 2$.
Find the area enclosed by the curves $y = x^2$ and $y = 2x - x^2$.
Note: The bounds are not given. Find where the curves intersect.
✅ Checkpoint: If you can solve Level 4 problems consistently, you’re ready for Area When Curves Cross and Integration with Respect to y.
Consider two parabolas: $y = x^2$ and $y = kx - x^2$ where $k > 0$.
(a) Find the intersection points of these curves in terms of $k$.
(b) Show that the area enclosed by the two parabolas is $\frac{k^3}{12}$.
(c) For what value of $k$ is the enclosed area equal to $\frac{8}{3}$?
CCI-Style Conceptual Questions
A student sets up the integral $\int_0^3 [g(x) - f(x)]\,dx$ to find the area between curves $f$ and $g$, where $f(x) > g(x)$ for all $x$ in $[0, 3]$. The student gets the answer $-12$.
What is the actual area of the region?
Common Misconceptions
reading a graph of two curves tells you which is “on top” without testing a point.
This is the iconic-graph error. The graph shows the curves visually, but which curve is the top boundary in the integral depends on the $y$-values at each $x$, not on which curve appears higher on the printed page when the scale is distorted or when the curves intersect. For the parabola $y = x^2$ and the line $y = 2x - x^2$ on $[0, 1]$, checking $x = 0.5$ gives $0.25$ versus $0.75$: only by evaluating both functions at the same $x$ can one confirm that $2x - x^2$ is the top curve throughout the interval.
Mastery Checklist
✅ If you check all boxes: You’re ready to tackle curves that cross and y-integration!
Exam Strategy Tips
🎯 How to approach area problems on exams
Step 0: Read carefully. Are bounds given? If not, you need intersection points.
Step 1: Quick sketch. Even a rough sketch prevents top/bottom errors. Label curves.
Step 2: Test a point. Pick an $x$-value between bounds. Which $y$ is larger? That’s the top.
Step 3: Set up the integral. Write $\int_a^b (\text{top} - \text{bottom})\,dx$ with explicit parentheses.
Step 4: Compute carefully. Most errors are algebraic. Distribute negatives properly.
Step 5: Sanity check.
- Is the answer positive? (It must be)
- Is it reasonable? (Think about the region’s size)
- Did you use the right bounds?
Time management: Finding intersections can be algebraically intensive. If you’re stuck, move on and return later.
Mental Model
The “Stacking Rectangles” Picture:
Imagine slicing the region into many thin vertical strips. Each strip is approximately a rectangle with:
- Height = (top curve) $-$ (bottom curve)
- Width = $dx$
The area integral “stacks” all these rectangles together, giving the total area.
Quick check: If your answer is negative, you subtracted in the wrong order. Take the absolute value.
Connections
Looking back:
- This extends the idea of “area under a curve” from Definite Integrals: we’re now finding area between two curves instead of between one curve and the x-axis
Looking ahead:
- Integration with respect to y handles regions that are easier to slice horizontally
- Volumes by disk/washer extends this idea to 3D: rotating the region gives a solid whose cross-sections are disks or washers
Real-world connections:
- Finding the area between a supply and demand curve gives consumer/producer surplus in economics
- The area between velocity curves of two cars gives how far apart they end up
- In epidemiology, area under a pathogenesis curve represents “amount of infection”
- The Gini Index measures income inequality using area between curves
🌍 Application: The Gini Index (Income Inequality)
The Gini Index is a real measure used by economists and governments worldwide to quantify income inequality. It’s computed using area between curves!
How it works:
- The Lorenz curve $y = L(x)$ plots cumulative income: if the poorest $a\%$ of households earn $b\%$ of total income, then $L(a/100) = b/100$.
- In a perfectly equal society, $L(x) = x$ (everyone earns the same).
- The Gini Index is defined as:
$$G = 2\int_0^1 [x - L(x)]\,dx$$
Real data (US, 2016): The poorest 40% of households received only 11.4% of total income, giving $L(0.4) = 0.114$. The Gini Index was approximately 0.48, indicating substantial inequality.
Why this matters: This is exactly the skill you’re learning: finding the area between the line $y = x$ and a curve below it. The same integral technique economists use to analyze inequality, you can now compute!
📚 Historical Note: The Area Problem
The problem of finding areas bounded by curves was one of the driving forces behind the development of calculus. Archimedes (287-212 BCE) computed areas using the “method of exhaustion,” essentially Riemann sums with geometric shapes.
The breakthrough came when Newton and Leibniz (independently, 1680s) discovered that area problems could be solved by finding antiderivatives, connecting two seemingly unrelated problems. This is why the Fundamental Theorem of Calculus is so profound: it says differentiation and integration are inverse operations.
The technique of finding area between curves is a natural extension that appears in Stewart’s textbook (and most calculus texts) because it reinforces the “slice and sum” intuition while preparing for volume problems.
Summary
| Concept | Key Point |
|---|---|
| Main formula | $A = \int_a^b [f(x) - g(x)]\,dx$ where $f$ is the top curve |
| Order matters | Always (top) − (bottom); negative result = wrong order |
| Finding bounds | If not given, solve $f(x) = g(x)$ for intersection points |
| Verification | Result should be positive and geometrically reasonable |
| When to use | When top/bottom boundaries are clear functions of $x$ |
| Next skill | When curves cross or when $y$-integration is easier |
| Previous | Up | Next |
|---|---|---|
| Section 5.1 | Section 5.1 | Integration with Respect to y |
Last updated: 2026-01-22