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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.

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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.

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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

🔄 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

✅ Checkpoint: If you can solve Level 4 problems consistently, you're ready for Area When Curves Cross and Integration with Respect to y.

CCI-Style Conceptual Questions

Mastery Checklist

• [ ] I can identify which curve is on top and which is on bottom for a given region
• [ ] I can set up the integral $\int_a^b [f(x) - g(x)]\,dx$ with the correct order of subtraction
• [ ] I can evaluate area integrals using the Fundamental Theorem of Calculus
• [ ] I understand why we subtract (top minus bottom) and what happens if we reverse the order
• [ ] I can handle problems where the bounds are given as well as problems where I must find them
• [ ] I can verify my answer is reasonable (positive, correct magnitude)

✅ If you check all boxes: You're ready to tackle curves that cross and y-integration!

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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 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.

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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:

1. The Lorenz curve $y = L(x)$ plots cumulative income: if the poorest $a\%$ of households earn $b\%$ of total income, then $L(a/100) = b/100$.
2. In a perfectly equal society, $L(x) = x$ (everyone earns the same).
3. 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.

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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

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Section 5.1	Section 5.1	Integration with Respect to y

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Last updated: 2026-01-22