Applications of Integrals

Area Between Two Curves

Formula: A = ∫[a to b] [f(x) - g(x)] dx

Where f(x) ≥ g(x) on [a, b]

Key: Integrate (top - bottom) or (right - left)

Example 1: Area Between Curves

Find area between y = x² and y = x from x = 0 to x = 1

Which is on top? At x = 0.5: x = 0.5, x² = 0.25 So x ≥ x² on [0, 1]

Calculate:

A = ∫[0 to 1] (x - x²) dx
  = [x²/2 - x³/3]₀¹
  = (1/2 - 1/3) - 0
  = 3/6 - 2/6
  = 1/6 square units

Example 2: Intersection Points

Find area between y = x² and y = 2x

Find intersections: x² = 2x → x² - 2x = 0 → x(x - 2) = 0 So x = 0 or x = 2

Which is on top? 2x ≥ x² on [0, 2]

Calculate:

A = ∫[0 to 2] (2x - x²) dx
  = [x² - x³/3]₀²
  = (4 - 8/3) - 0
  = 12/3 - 8/3
  = 4/3 square units

Volumes of Revolution: Disk Method

Rotate region around axis

Formula (around x-axis):

V = π∫[a to b] [f(x)]² dx

Geometric idea: Stack circular disks

Example: Disk Method

Rotate y = √x from x = 0 to x = 4 around x-axis

Calculate:

V = π∫[0 to 4] (√x)² dx
  = π∫[0 to 4] x dx
  = π[x²/2]₀⁴
  = π(8 - 0)
  = 8π cubic units

Volumes: Washer Method

Region between two curves rotated

Formula: V = π∫[a to b] [R(x)² - r(x)²] dx

Where R(x) = outer radius, r(x) = inner radius

Creates washers (disks with holes)

Example: Washer Method

Rotate region between y = x and y = x² around x-axis (from x = 0 to x = 1)

Outer radius: R = x Inner radius: r = x²

Calculate:

V = π∫[0 to 1] (x² - x⁴) dx
  = π[x³/3 - x⁵/5]₀¹
  = π(1/3 - 1/5)
  = π(5/15 - 3/15)
  = 2π/15 cubic units

Volumes: Shell Method

Rotate around axis using cylindrical shells

Formula (rotate around y-axis):

V = 2π∫[a to b] x·f(x) dx

Useful when disk/washer method is complicated

Example: Shell Method

Rotate y = x² from x = 0 to x = 2 around y-axis

Using shells:

V = 2π∫[0 to 2] x·x² dx
  = 2π∫[0 to 2] x³ dx
  = 2π[x⁴/4]₀²
  = 2π(16/4)
  = 8π cubic units

Arc Length

Length of curve from x = a to x = b

Formula: L = ∫[a to b] √(1 + [f'(x)]²) dx

Derived from Pythagorean theorem on infinitesimal segments

Example: Arc Length

Find length of y = x^(3/2) from x = 0 to x = 1

Find derivative: y' = (3/2)x^(1/2)

Calculate:

L = ∫[0 to 1] √(1 + (9/4)x) dx

Let u = 1 + (9/4)x, du = (9/4)dx:

= (4/9)∫[1 to 13/4] √u du
= (4/9) · (2/3)u^(3/2)|₁^(13/4)
= (8/27)[(13/4)^(3/2) - 1]
≈ 1.44 units

Surface Area of Revolution

Rotate curve around axis

Formula (around x-axis):

S = 2π∫[a to b] f(x)√(1 + [f'(x)]²) dx

Like arc length with radius factor

Example: Surface Area

Rotate y = x from x = 0 to x = 1 around x-axis (cone)

y' = 1

Calculate:

S = 2π∫[0 to 1] x√(1 + 1) dx
  = 2π√2 ∫[0 to 1] x dx
  = 2π√2 [x²/2]₀¹
  = π√2 square units

Work Done by Variable Force

Work = ∫[a to b] F(x) dx

F(x) = force as function of position

Example: Spring Work

Hooke's Law: F = kx (k = spring constant)

Compress spring 0.2 m, k = 100 N/m. Work done?

Calculate:

W = ∫[0 to 0.2] 100x dx
  = 100[x²/2]₀^0.2
  = 50(0.04)
  = 2 joules

Pumping Liquids

Work to pump liquid out of tank

Consider: Weight, distance each slice must travel

Example: Cylindrical Tank

Cylinder: radius 2 m, height 5 m, full of water

Pump water to top. Water density: 1000 kg/m³

Slice at height y has:

• Volume: π(2)²·dy = 4π dy
• Weight: 1000g(4π dy) ≈ 40000π dy N
• Distance to top: (5 - y) m

Work:

W = ∫[0 to 5] 40000π(5 - y) dy
  = 40000π[5y - y²/2]₀⁵
  = 40000π[25 - 12.5]
  = 500000π joules

Average Value of Function

Average over [a, b]:

f_avg = (1/(b-a))∫[a to b] f(x) dx

Geometric: Height of rectangle with same area

Example: Average Temperature

Temperature T(t) = 60 + 20sin(πt/12) over 24 hours

Average:

T_avg = (1/24)∫[0 to 24] (60 + 20sin(πt/12)) dt

Evaluate:

= (1/24)[60t - (240/π)cos(πt/12)]₀^24
= (1/24)[1440 - 0 - (0 - 240/π)]
= 60°F

(Sine averages to 0 over full period)

Physics: Center of Mass

1D center of mass:

x̄ = (∫[a to b] x·ρ(x) dx)/(∫[a to b] ρ(x) dx)

Where ρ(x) = density function

Economics: Consumer Surplus

Difference between what consumers willing to pay and actually pay

CS = ∫[0 to Q] [D(q) - P] dq**

D(q) = demand function, P = equilibrium price*

Probability: Continuous Distributions

Probability density function f(x)

P(a < X < b) = ∫[a to b] f(x) dx

Total probability: ∫[-∞ to ∞] f(x) dx = 1

Example: Uniform Distribution

f(x) = 1/10 on [0, 10]

P(2 < X < 5)?

= ∫[2 to 5] (1/10) dx
= (1/10)[x]₂⁵
= (1/10)(3)
= 0.3 or 30%

Biology: Population Growth

Logistic growth: dP/dt = rP(1 - P/K)

Total population change:

ΔP = ∫[t₁ to t₂] r P(1 - P/K) dt

Practice