Chapter 28 — Applications of Integration

Area, Volume, Work, and Centers of Mass

28.1 Learning Objectives

By the end of this chapter, students should be able to:

• Apply integration to real physical and geometric problems.
• Compute areas between curves.
• Compute volumes of solids.

Understand:

• disk method

washer method

shell method

Interpret physical work mathematically.

Understand density and center of mass conceptually.

Apply integrals to accumulation of physical quantities.

Visualize three-dimensional accumulation processes.

Understand why integration became foundational in physics and engineering.

28.2 Big Picture — Integration Measures Accumulated Reality

Earlier chapters developed:

integration techniques

accumulation concepts

definite integrals

Now calculus applies these ideas to:

geometry

physics

engineering

physical systems

Integration becomes:

a universal accumulation tool.

This chapter reveals one of calculus’s deepest ideas:

many complex physical quantities arise from adding infinitely many tiny contributions.

28.3 From Area to Physical Quantities

At first:

integrals measured area.

But the same accumulation principle applies to:

volume

mass

force

energy

probability

fluid flow

charge distribution

Integration becomes:

the mathematics of continuous accumulation.

28.4 Area Between Curves

Earlier:

area measured between curve and x-axis.

Now we measure area between:

two curves.

Suppose:

upper curve:

f(x)

and lower curve:

g(x)

Area formula:

∫

a

b

​

[f(x)−g(x)]dx

28.5 Why Subtraction Appears

Each vertical slice has height:

f(x)−g(x)

The integral accumulates:

infinitely many tiny vertical strips.

Students should visualize:

stacked slices filling region.

28.6 Worked Example — Area Between Curves

Find area between:

y = x

and:

y = x

2

from:

x =0

to:

x =1

Upper curve:

x

Lower curve:

x

2

Area:

∫

0

1

​

(x−x

2

)dx

28.7 Evaluate the Integral

Integrate:

2

x

2

​

−

3

x

3

​

Evaluate from:

0

to:

1

Result:

2

1

​

−

3

1

​

=

6

1

​

Area equals:

6

1

​

square units.

28.8 Why This Is Important Geometrically

Integration allows exact measurement of:

irregular curved regions

that ordinary geometry cannot handle easily.

This transformed geometry historically.

28.9 Volumes of Solids — Extending Area into 3D

Now calculus studies:

three-dimensional accumulation.

Suppose region rotates around axis.

The rotation creates:

solid object.

Integration accumulates:

infinitely many thin cross-sections

to produce:

exact volume.

28.10 Disk Method

Suppose cross-sections form:

solid disks.

Disk volume:

πr

2

Accumulating infinitely many disks:

V =∫

a

b

​

π[r(x)]

2

dx

28.11 Why Disks Naturally Appear

Rotating curves around axes naturally produces:

circular slices.

Students should visualize:

stacked coins

layered cylinders

growing solids

Integration accumulates:

infinitely many tiny disks.

28.12 Worked Example — Disk Method

Rotate:

y = x

from:

0

to:

1

around x-axis.

Radius:

r = x

Volume:

V =∫

0

1

​

πx

2

dx

Integrate:

π[

3

x

3

​

]

0

1

​

Result:

3

π

​

28.13 Washer Method — Hollow Solids

Suppose solid contains:

hollow center.

Cross-sections become:

washers

with:

outer radius

inner radius

Volume formula:

V =∫

a

b

​

π(R

2

−r

2

)dx

where:

R = outer radius

r = inner radius

28.14 Why Subtraction Appears Again

Outer disk contains:

unwanted inner hole.

Subtract inner volume from outer volume.

Students should visualize:

hollow rings stacking together.

28.15 Shell Method — Cylindrical Layers

Sometimes rotating vertical strips becomes awkward.

Instead:

rotate thin strips into cylindrical shells.

Shell volume:

2π(radius)(height)(thickness)

Formula:

V =∫

a

b

​

2π(radius)(height)dx

28.16 Why Multiple Volume Methods Exist

Different geometries favor:

different slicing strategies.

Students should increasingly ask:

what structure simplifies accumulation?

This strategic thinking becomes central in advanced calculus.

28.17 Physical Work — Accumulating Force

Physics defines work as:

W = Fd

when force constant.

But many systems possess:

changing force.

Examples:

springs

gravity

fluid pressure

Integration accumulates:

infinitely many tiny force contributions.

28.18 Work Integral

Suppose:

force varies with position.

Tiny work contribution:

dW = F(x)dx

Total work:

W =∫

a

b

​

F(x)dx

Integration accumulates:

tiny energy transfers.

28.19 Worked Example — Spring Force

Hooke’s Law:

F(x)= kx

Suppose:

k =4

Stretch spring from:

0

to:

3

Work:

∫

0

3

​

4xdx

Integrate:

2x

2

Evaluate:

18

units of work.

28.20 Why Work Integrals Matter

Work integrals became foundational in:

mechanics

thermodynamics

electromagnetism

engineering

Integration naturally models:

accumulated energy transfer.

28.21 Centers of Mass

Real objects possess:

distributed mass.

Question:

Where does object balance?

Integration computes:

weighted average position.

This leads to:

center of mass.

28.22 Why Simple Averaging Fails

Irregular objects possess:

uneven density

varying geometry

Ordinary averages insufficient.

Integration accumulates:

infinitely many tiny weighted contributions.

28.23 Basic Center of Mass Idea

Moment:

(mass)(distance)

Center of mass balances:

total moments.

Formula structure:

x

ˉ

=

total mass

total moment

​

Integration computes both quantities continuously.

28.24 Density Functions

Objects may possess:

varying density.

Example:

thicker regions

heavier materials

changing composition

Density function:

ρ(x)

Mass integral:

m =∫

a

b

​

ρ(x)dx

28.25 Why Centers of Mass Matter Physically

Applications:

engineering design

aircraft stability

structural analysis

robotics

orbital mechanics

Balancing distributed mass became essential throughout physics.

28.26 Why This Chapter Is Important Historically

Applications of integration transformed:

engineering

architecture

mechanics

astronomy

manufacturing

Integration became practical mathematics for:

the physical world.

28.27 Relationship to Earlier Calculus Ideas

This chapter unifies:

accumulation

geometry

physics

limits

definite integrals

continuous systems

Students now see integration as:

universal accumulation machinery.

28.28 Why Infinitely Many Tiny Pieces Matter

Area:

tiny rectangles

Volume:

tiny disks/shells

Work:

tiny force contributions

Mass:

tiny density contributions

The same accumulation principle appears everywhere.

This is one of calculus’s deepest unifying ideas.

28.29 Common Student Mistakes

Mistake 1 — Using Wrong Radius

Volume problems require careful geometry.

Mistake 2 — Forgetting Outer Minus Inner

Washer method requires subtraction.

Mistake 3 — Losing Physical Interpretation

Integrals represent:

accumulated quantities

not merely formulas.

Mistake 4 — Confusing Shell and Disk Methods

Different slicing approaches produce:

different formulas.

28.30 Visualization Strategy

Students should continually imagine:

thin slices

rotating regions

stacked disks

cylindrical shells

accumulating force

balancing mass

This chapter is deeply geometric and physical.

28.31 Why This Chapter Matters

This chapter reveals:

integration as universal accumulation mathematics.

Students now see calculus directly modeling:

physical reality

geometry

engineering systems

continuous natural processes.

28.32 Practice Problems

A. Area Between Curves

Find area between:

y = x

and:

y = x

2

on:

[0,1]

Explain why subtraction appears.

Explain why upper minus lower matters.

Explain why area between curves generalizes ordinary area.

Explain why integration handles curved regions naturally.

B. Disk and Washer Methods

Find volume generated by rotating:

y = x

from:

0

to:

1

around x-axis.

Explain why disks appear naturally.

Explain washer method geometrically.

Explain why hollow regions require subtraction.

Explain why volume becomes accumulated area.

C. Shell Method

Explain shell method conceptually.

Explain why cylindrical shells arise.

Explain difference between:

shell method

disk method

Explain why different slicing strategies exist.

Explain why geometry determines integration strategy.

D. Work Integrals

Compute work for:

F(x)=3x

from:

0

to:

2

Explain why varying force requires integration.

Explain why work accumulates continuously.

Explain why energy transfer appears naturally in integrals.

Explain why work became important in physics.

E. Centers of Mass

Explain what center of mass means physically.

Explain why distributed mass requires integration.

Explain why density functions matter.

Explain why weighted averages appear.

Explain why balancing systems require moments.

F. Conceptual Problems

Explain why integration became essential in engineering.

Explain relationship between:

geometry

accumulation

physics

Explain why infinitely many tiny pieces produce physical totals.

Explain why applications of integration transformed science.

Explain why this chapter reveals the true power of calculus.

28.33 Selected Solutions

Problem 1

Area:

∫

0

1

​

(x−x

2

)dx

Integrate:

[

2

x

2

​

−

3

x

3

​

]

0

1

​

Result:

6

1

​

Problem 6

Volume:

V =∫

0

1

​

πx

2

dx

Integrate:

π[

3

x

3

​

]

0

1

​

Result:

3

π

​

Problem 16

Work:

∫

0

2

​

3xdx

Integrate:

2

3

​

x

2

Evaluate:

6

units of work.

Problem 21

Center of mass represents:

balancing point of distributed mass

where total rotational effects balance equally.

28.34 Chapter Summary

In this chapter we introduced:

area between curves

disk method

washer method

shell method

work integrals

centers of mass

density functions

physical accumulation models

Most importantly:

students learned that integration serves as a universal mathematical framework for accumulating:

geometry

mass

energy

force

physical quantities

through infinitely many tiny contributions.