Section 5.1: Areas Between Curves
Course
MATH162
Overview
This section extends the definite integral from finding the area under a single curve to finding the area between two curves. The key insight: if you know how to find area under a curve, you can find area between curves by subtraction.
The Big Picture: Every area problem in this section reduces to one formula:
\[\boxed{A = \int_a^b \vert f(x) - g(x)\vert \, dx}\]The challenge is setting it up correctly—identifying which curve is “on top,” finding bounds, and choosing whether to integrate with respect to $x$ or $y$.
Skill Dependency Map
graph TD
subgraph Phase1["Phase 1: Foundations"]
A["Area Between Curves<br/>(Vertical Rectangles)"]
end
subgraph Phase2["Phase 2: Variations"]
B["Area Between Curves<br/>(Horizontal Rectangles)"]
C["Finding Intersection<br/>Points"]
end
subgraph Phase3["Phase 3: Complications"]
D["Area When<br/>Curves Cross"]
end
A --> B
A --> C
C --> D
B --> D
style A fill:#d1fae5,stroke:#10b981,stroke-width:2px
style B fill:#d1fae5,stroke:#10b981,stroke-width:2px
style C fill:#d1fae5,stroke:#10b981,stroke-width:2px
style D fill:#d1fae5,stroke:#10b981,stroke-width:2px
click A "../../skills/ch5-sec1/area-between-curves-vertical.html"
click B "../../skills/ch5-sec1/area-between-curves-horizontal.html"
click C "../../skills/ch5-sec1/finding-intersection-points.html"
click D "../../skills/ch5-sec1/area-curves-that-cross.html"
Learning Path
Phase 1: The Basic Setup
| Skill | Key Takeaway | Time |
|---|---|---|
| Area Between Curves (Vertical) | $A = \int_a^b (\text{top} - \text{bottom})\,dx$ | ~20 min |
Mastery check: Can you find the area between $y = x^2$ and $y = x + 2$?
Phase 2: Tools & Variations
| Skill | Key Takeaway | Time |
|---|---|---|
| Finding Intersection Points | Set $f(x) = g(x)$ and solve | ~15 min |
| Area Between Curves (Horizontal) | $A = \int_c^d (\text{right} - \text{left})\,dy$ | ~20 min |
Mastery check: Can you set up the area between $x = y^2$ and $x = y + 2$ using horizontal slices?
Phase 3: Handling Complications
| Skill | Key Takeaway | Time |
|---|---|---|
| Area When Curves Cross | Split at crossings; add absolute areas | ~20 min |
Mastery check: Can you find the area between $\sin x$ and $\cos x$ from $0$ to $\pi/2$?
Key Equations
| Formula | When to Use |
|---|---|
| $A = \int_a^b [f(x) - g(x)]\,dx$ | $f(x) \geq g(x)$ throughout $[a,b]$ |
| $A = \int_c^d [f(y) - g(y)]\,dy$ | Integrating with respect to $y$; $f(y) \geq g(y)$ |
| $A = \int_a^b \lvert f(x) - g(x) \rvert\,dx$ | Curves may cross; gives total area |
Memory aid: Always subtract in the direction perpendicular to your slices:
- Vertical slices → (top $-$ bottom) $dx$
- Horizontal slices → (right $-$ left) $dy$
Exercise Coverage Map
| Stewart Exercise | Skill(s) Needed | Difficulty |
|---|---|---|
| 1-4 | Vertical Area Setup | Basic |
| 5-6 | Vertical Area, Intersection Points | Basic |
| 7-10 | Setup only (no evaluation) | Intermediate |
| 11-18 | Choosing dx vs dy, Intersection Points | Intermediate |
| 19-34 | Full area calculations | Intermediate |
| 35 | Curves That Cross, Net vs Total | Conceptual |
| 39-40 | Triangles via Calculus | Application |
| 53-55 | Midpoint Rule applications | Numerical |
| 61-67 | Advanced: parameter problems | Challenge |
Self-Assessment Quiz
Test your understanding before moving to Section 5.2.
Q1: What integral gives the area between y = 4 and y = x²?
Answer: The curves intersect where $x^2 = 4$, so $x = \pm 2$.
\[A = \int_{-2}^{2} (4 - x^2)\,dx\]Since $y = 4$ is above $y = x^2$ on this interval, we compute (top $-$ bottom).
Q2: When should you integrate with respect to y instead of x?
Answer: Use $y$-integration when:
- The curves are given as $x = f(y)$ (like $x = y^2$)
- The left/right boundaries are simpler than top/bottom
- Using $x$-integration would require multiple integrals
Example: The area between $x = y^2$ and $x = y + 2$ is much easier with horizontal slices.
Q3: What’s wrong with ∫₀^π (cos x - sin x) dx for the area between sine and cosine?
Answer: The curves cross at $x = \pi/4$! Before this point, $\cos x > \sin x$, but after, $\sin x > \cos x$. The integral gives net signed area (positive minus negative), not total area.
Correct setup: \(A = \int_0^{\pi/4} (\cos x - \sin x)\,dx + \int_{\pi/4}^{\pi} (\sin x - \cos x)\,dx\)
Q4: The line y = x - 1 and parabola y² = 2x + 6 enclose a region. Which variable should you integrate with respect to?
Answer: Integrate with respect to $y$.
Express both as $x = f(y)$:
- Line: $x = y + 1$
- Parabola: $x = \frac{1}{2}y^2 - 3$
This gives one clean integral. Using $x$ would require splitting the region because the bottom boundary changes.
Key Mathematical Themes
| Theme | How It Appears in §5.1 |
|---|---|
| Subtraction gives difference | Area between = Area under top $-$ Area under bottom |
| Absolute value for total | $\lvert f - g \rvert$ ensures positive area even when curves cross |
| Choose your variable wisely | $dx$ vs $dy$ can turn 2 integrals into 1 |
| Geometry informs algebra | Sketch first to identify top/bottom, left/right |
| Intersection = boundary | Solve $f = g$ to find integration limits |
Deep Connections
Why This Section Matters
The area-between-curves formula is your first glimpse of a powerful pattern: integration measures accumulation of differences. This same structure appears in:
- Volumes (§5.2-5.3): Cross-sectional area varies with position
- Work (§5.4): Force varies with distance
- Probability: Area under a density curve between two values
- Economics: Consumer/producer surplus = area between supply and demand curves
The Geometric Viewpoint
When you compute $\int_a^b [f(x) - g(x)]\,dx$, you’re stacking infinitely thin vertical rectangles, each with height $f(x) - g(x)$ and width $dx$. This “stacking rectangles” picture will return when we compute volumes by slicing.
Connection to Definite Integrals
This section shows that $\int_a^b f(x)\,dx$ isn’t really “area under the curve”—it’s signed area between the curve and the $x$-axis. The “area between curves” formula just replaces the $x$-axis with another curve.
Common Mistakes to Avoid
| Mistake | Why It’s Wrong | Fix |
|---|---|---|
| Always subtracting in given order | Negative area if order is wrong | Always compute (top $-$ bottom) or (right $-$ left) |
| Forgetting to find intersections | Wrong bounds give wrong answer | When curves “enclose” a region, solve $f = g$ |
| Not checking for crossings | Net area $\neq$ total area | Test a point in each subinterval |
| Forcing $x$-integration | May require multiple integrals | Check if $y$-integration is simpler |
Skills in This Section
| Skill | Description | Difficulty |
|---|---|---|
| Area Between Curves (Horizontal Rectangles) | Applications of Integration | Intermediate |
| Area Between Curves (Integrating with Respect to x) | Applications of Integration | Beginner |
| Area Between Curves (Integrating with Respect to y) | Applications of Integration | Intermediate |
| Area Between Curves (Vertical Rectangles) | Applications of Integration | Beginner |
| Area Between Curves That Cross | Applications of Integration | Intermediate |
| Finding Intersection Points for Area Problems | Applications of Integration | Intermediate |
| Previous | Up | Next |
|---|---|---|
| Chapter 5 | Chapter 5 | Section 5.2: Volumes |