Geometric Applications & Trigonometric Substitutions

Arc Length Formula

L = ∫[a to b] √(1 + [f'(x)]²) dx

Volume: Disk Method

V = ∫[a to b] π(Radius)² dx = ∫[a to b] π[R(x)]² dx

Volume: Washer Method

V = ∫[a to b] π[(Outer Radius)² - (Inner Radius)²] dx

Volume: Shell Method

V = ∫[a to b] 2π(Shell Radius)(Shell Height) dx

Trig Sub for √(a² - x²)

x = a * sin θ , dx = a * cos θ dθ , Expression = a * cos θ

Trig Sub for √(a² + x²)

x = a * tan θ , dx = a * sec² θ dθ , Expression = a * sec θ

Trig Sub for √(x² - a²)

x = a * sec θ , dx = a * sec θ tan θ dθ , Expression = a * tan θ