Calculus II · Core Concepts Quiz

1

Integration Techniques

Integration by Parts: ∫u dv = uv − ∫v du
Trigonometric Substitution:
√(a²−x²) → x = a sinθ
√(a²+x²) → x = a tanθ
√(x²−a²) → x = a secθ
Partial Fractions: decompose rational functions

Example

∫x eˣ dx — use IBP: u=x, dv=eˣdx ⟹ uv−∫v du = xeˣ − eˣ + C

💡 LIATE rule for choosing u: Logarithm, Inverse trig, Algebraic, Trig, Exponential

2

Improper Integrals & Applications

∫₁^∞ 1/xᵖ dx converges iff p > 1
Arc Length: L = ∫ₐᵇ √(1 + [f′(x)]²) dx
Surface Area: S = 2π ∫ₐᵇ f(x) √(1 + [f′(x)]²) dx
Volume (disk): V = π ∫ₐᵇ [f(x)]² dx

Example

∫₁^∞ 1/x² dx = lim[b→∞] [−1/x]₁ᵇ = 0 − (−1) = 1 → Converges

3

Sequences & Series

Geometric: Σ arⁿ converges to a/(1−r) if |r|<1
p-Series: Σ 1/nᵖ converges iff p > 1
Ratio Test: L = lim|aₙ₊₁/aₙ|; converges if L<1
Alternating Series Test: converges if bₙ↓0
Power Series radius: use Ratio Test on aₙ

Example

Σ (1/2)ⁿ = 1/(1−1/2) = 2 (geometric, a=1, r=1/2)

💡 Common Maclaurin series:
eˣ = Σ xⁿ/n! | sin x = Σ (−1)ⁿx^(2n+1)/(2n+1)! | cos x = Σ (−1)ⁿx^(2n)/(2n)! | 1/(1−x) = Σ xⁿ

4

Parametric Equations & Polar Coordinates

dy/dx = (dy/dt)/(dx/dt)
Arc Length (param): L = ∫ₐᵇ √((dx/dt)² + (dy/dt)²) dt
Polar Area: A = ½ ∫ₐᵇ r² dθ
Polar Arc Length: L = ∫ₐᵇ √(r² + (dr/dθ)²) dθ

Example

x=t², y=t³: dy/dx = (3t²)/(2t) = 3t/2

5

Taylor & Maclaurin Series

Taylor: f(x) = Σ f⁽ⁿ⁾(a)/n! · (x−a)ⁿ
Maclaurin (a=0): f(x) = Σ f⁽ⁿ⁾(0)/n! · xⁿ
Error bound (alt series): |Error| ≤ |aₙ₊₁|
Radius of convergence R: via Ratio or Root Test

Example

Maclaurin for eˣ: e² ≈ 1 + 2 + 4/2! + 8/3! = 1+2+2+4/3 ≈ 6.33