📐 Concept diagram

05-09 - Applications of Integration

Phase: 5 | Subject: 05-09 Prerequisites: 05-02-the-definite-integral.md, 05-05-integration-by-parts.md Next subject: 05-10-parametric-equations-and-polar-coordinates.md

Learning Objectives

By the end of this subject, you will be able to:

1. Calculate area between two curves
2. Calculate volume by disks and washers
3. Calculate volume by cylindrical shells
4. Calculate arc length
5. Calculate surface area of revolution
6. Calculate average value of a function

Core Content

Area Between Two Curves

$Area = ∫[a to b] [top(x) - bottom(x)] dx
$

where the curves intersect at x = a and x = b.

Example: Area between y = x² and y = x from x = 0 to x = 1. Top: y = x, Bottom: y = x² Area = ∫[0 to 1] (x - x²)dx = [x²/2 - x³/3][0 to 1] = 1/2 - 1/3 = 1/6

Volume by Disks

When rotating y = f(x) around the x-axis:

$V = π ∫[a to b] [f(x)]² dx
$

Example: Rotate y = √x from x = 0 to x = 4 around x-axis. V = π ∫[0 to 4] x dx = π[x²/2][0 to 4] = π(8) = 8π

Volume by Washers

When rotating the region between two curves around an axis:

$V = π ∫[a to b] ([outer]² - [inner]²) dx
$

Volume by Shells

When rotating around the y-axis:

$V = 2π ∫[a to b] x·f(x) dx
$

When to use shells: When the region is easier to describe as x = f(y) or when rotating around y-axis with y = f(x).

Arc Length

$L = ∫[a to b] √(1 + (dy/dx)²) dx
$

Example: Length of y = x² from x = 0 to x = 1. dy/dx = 2x. L = ∫[0 to 1] √(1 + 4x²)dx. This requires trig substitution — result involves ln and √5.

Surface Area of Revolution

Around x-axis:

$S = 2π ∫[a to b] y·√(1 + (dy/dx)²) dx
$

Average Value

$f_avg = (1/(b-a)) ∫[a to b] f(x)dx
$

Mean Value Theorem for Integrals: There exists c in [a, b] such that f(c) = f_avg.

Key Terms

• 05 09 Applications Of Integration
• Arc Length
• Area Between Two Curves
• Average Value
• Correct: A)
• Correct: B)
• Correct: C)
• Example 1: Area between curves
• Example 2: Volume by washers
• Example 3: Average value
• Surface Area of Revolution
• Volume by Disks

Worked Examples

Example 1: Area between curves

y = x² and y = 2x. Intersection: x² = 2x → x = 0, 2. Top: y = 2x, Bottom: y = x². Area = ∫[0 to 2] (2x - x²)dx = [x² - x³/3][0 to 2] = 4 - 8/3 = 4/3

Example 2: Volume by washers

Rotate region between y = x and y = x² from x = 0 to 1 around x-axis. Outer: y = x, Inner: y = x². V = π ∫[0 to 1] (x² - x⁴)dx = π[x³/3 - x⁵/5][0 to 1] = π(1/3 - 1/5) = 2π/15

Example 3: Average value

Average of f(x) = x² on [0, 2]. f_avg = (1/2) ∫[0 to 2] x²dx = (1/2)[x³/3][0 to 2] = (1/2)(8/3) = 4/3

Practice Problems

Problem 1: Area between y = x and y = x² from 0 to 1

Problem 2: Volume of rotating y = √x, 0 ≤ x ≤ 4, around x-axis

Problem 3: Average value of f(x) = sin(x) on [0, π]

Problem 4: Arc length of y = (2/3)x^(3/2) from x = 0 to x = 3

Problem 5: Volume: rotate region bounded by y = x², x = 0, x = 2, y = 0 around y-axis using shells

Summary

Key takeaways:

• Area between curves: ∫(top - bottom)
• Disk method: V = π∫y²dx (rotation around x-axis)
• Washer: V = π∫(outer² - inner²)
• Shell: V = 2π∫x·y dx (rotation around y-axis)
• Arc length: ∫√(1 + (dy/dx)²) dx
• Average value: (1/(b-a)) ∫[a to b] f(x)dx

Pitfalls

• Choosing the wrong curve as top/bottom. Area between curves is ∫(top − bottom). If the curves cross within the interval, you must split at the intersection point(s). Assuming one curve is always above the other without checking is a common error.
• Confusing the disk and shell methods. Disk/washer methods integrate perpendicular to the axis of rotation (cross-sectional area). The shell method integrates parallel to the axis (cylindrical shells). Using the wrong method leads to integrals that are impossible to set up or have the wrong variable of integration.
• Forgetting to square in the disk formula. V = π∫[f(x)]²dx. The square comes from the area of a circular cross-section: πr². Writing π∫f(x)dx instead misses the square and gives a completely wrong (and dimensionally inconsistent) result.
• Using the wrong radius in the shell method. When rotating around the y-axis using shells, the radius is x (the horizontal distance from the axis of rotation to the shell). Using f(x) or y as the radius is incorrect.
• Mishandling the arc length formula under the radical. L = ∫√(1 + (dy/dx)²)dx. Students sometimes write √((dy/dx)²) = |dy/dx| (dropping the 1) or misinterpret (1 + dy/dx)². The expression under the radical is 1 plus the square of the derivative, not the square of (1 + dy/dx).

Quiz

Q1: Area between y = x and y = x² from x = 0 to 1:

A) 1/6 B) 1/3 C) 1/2 D) 1/12

Q2: Volume by rotating y = x² from 0 to 2 around x-axis:

A) 8π/3 B) 16π/3 C) 32π/5 D) 8π

Q3: Average value of f(x) = x² on [0, 2]:

A) 4/3 B) 2 C) 4 D) 8/3

Q4: Volume by shells for rotating y = x², 0 ≤ x ≤ 1, around y-axis:

A) π/2 B) π/3 C) π/4 D) π/5

Q5: The average value of f(x) on [a, b] is:

A) (f(a) + f(b))/2 B) (1/(b-a)) ∫[a to b] f(x)dx C) ∫[a to b] f(x)dx D) f((a+b)/2)

Next Steps

Next up: 05-10-parametric-equations-and-polar-coordinates.md