Chapter 5: Applications of Integration

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The Big Picture

You’ve learned to compute definite integrals. Now we ask: what can you do with them?

This chapter reveals a powerful pattern: whenever a quantity can be approximated by a Riemann sum, its exact value is a definite integral. This single idea—slice, approximate, sum, take limits—handles an astonishing range of problems:

The underlying strategy is always the same: break the problem into infinitely many tiny pieces, each of which you can handle, then add them all up.

Chapter Flow

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        A["§5.1 Areas<br/>Between Curves"]
        B["§5.2 Volumes<br/>(Disk/Washer)"]
        C["§5.3 Volumes<br/>(Shells)"]
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        D["§5.4 Work"]
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    subgraph "Analysis"
        E["§5.5 Average<br/>Value"]
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Sections

Section 5.1: Areas Between Curves

Core formula: $A = \int_a^b [\text{top} - \text{bottom}]\,dx$

Find the area of a region bounded by two curves. Choose whether to integrate with respect to $x$ or $y$ based on which is simpler.

Section 5.2: Volumes

Core formula: $V = \int_a^b A(x)\,dx$

Compute volumes by slicing a solid into thin cross-sections. Special cases include:

• Disk method: $A(x) = \pi r^2$ when rotating a region about an axis it touches
• Washer method: $A(x) = \pi(R^2 - r^2)$ when rotating a region about an axis it doesn’t touch

Section 5.3: Volumes by Cylindrical Shells

Core formula: $V = \int_a^b 2\pi rh\,dx$

An alternative to disks/washers: slice parallel to the axis of rotation, creating cylindrical shells. Often simpler when solving for $x$ would be difficult.

Section 5.4: Work

Core formula: $W = \int_a^b F(x)\,dx$

Compute the energy required to move an object when force varies with position. Classic applications include springs (Hooke’s Law), cables/chains, and pumping water.

Section 5.5: Average Value of a Function

Core formula: $f_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dx$

Find the “typical” value of a continuous function over an interval. The Mean Value Theorem for Integrals guarantees that $f$ actually achieves this average value at some point.

The Unifying Theme

Every formula in this chapter follows the same pattern:

Once you see this pattern, you can apply it to new situations beyond the textbook.

Prerequisites

Before starting Chapter 5, ensure you can:

• Compute definite integrals using the Fundamental Theorem of Calculus
• Apply u-substitution to evaluate integrals
• Find where two curves intersect (solve $f(x) = g(x)$)
• Sketch graphs of polynomial, exponential, and trigonometric functions
• Set up and interpret Riemann sums

Review: Chapter 4: Integrals

Study Tips

1. Draw pictures. Every problem in this chapter has a geometric interpretation. Sketch before you set up the integral.

2. Identify what varies. Ask: what quantity changes as I move through the region or solid? That’s your variable of integration.

3. Check dimensions. Area integrals should give square units; volume integrals give cubic units; work gives energy units (joules or ft-lb).

4. Compare methods. Often there’s more than one way to set up a problem. Try both and see which is simpler.

5. Verify with special cases. Does your volume formula give $\frac{4}{3}\pi r^3$ for a sphere? Does your work formula give $Fd$ when force is constant?