Calc 2 - Complete Study Guide

Integration by Parts

∫u dv = uv - ∫v du

1. LIATE Priority: Logs → Inverse trig → Algebraic → Trig → Exponential

2. When to Use: Product of two different function types (xe^x, x sin x, x ln x)

3. Sometimes Apply Twice: For x²e^x or x² sin x, use parts twice in a row

Trig Integrals and Substitution

Essential Trig Identities:

sin²x + cos²x = 1

tan²x + 1 = sec²x

1 + cot²x = csc²x

sin²x = (1 - cos 2x)/2

cos²x = (1 + cos 2x)/2

1. Odd Powers of Sine/Cosine: Save one factor, convert rest using sin²x + cos²x = 1

2. Even Powers: Use half-angle formulas to reduce power

3. Trig Substitution Patterns:

√(a² - x²) → x = a sin θ

√(a² + x²) → x = a tan θ

√(x² - a²) → x = a sec θ

Partial Fractions

1. Factor Denominator Completely: Find all linear and irreducible quadratic factors

2. Set Up Fractions:

Linear factor (x - a): A/(x - a)

Repeated linear (x - a)²: A/(x - a) + B/(x - a)²

Irreducible quadratic (x² + bx + c): (Ax + B)/(x² + bx + c)

3. Finding Coefficients:

Method 1: Multiply through and equate coefficients

Method 2: Substitute convenient x-values to solve for constants

Method 3: Cover-up method for simple linear factors

Approximate Integrals

Trapezoidal Rule: T_n = (Δx/2)[f(x₀) + 2f(x₁) + 2f(x₂) + ... + 2f(x_{n-1}) + f(x_n)]

Simpson's Rule: S_n = (Δx/3)[f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + ... + f(x_n)]

1. Δx = (b - a)/n where n is number of subintervals

2. Simpson's Rule: Requires even number of subintervals, more accurate

3. Error Bounds: Simpson's error goes as (Δx)⁴, Trapezoidal as (Δx)²

Improper Integrals

∫[a to ∞] f(x) dx = lim[t→∞] ∫[a to t] f(x) dx

1. Type 1: Infinite limits - take limit as you approach infinity

2. Type 2: Discontinuous integrand - take limit as you approach discontinuity

3. Convergence vs Divergence:

Converges: Limit exists and is finite

Diverges: Limit is infinite or doesn't exist

Compare with p-integrals: ∫[1 to ∞] 1/x^p converges if p > 1

Areas Between Curves

A = ∫[a to b] |f(x) - g(x)| dx

1. Always Sketch: Determine which function is on top in each interval

2. Find Intersection Points: Solve f(x) = g(x) to find limits of integration

3. Split at Crossings: If curves cross, split into separate integrals

Volumes

Disk Method: V = π∫[a to b] [R(x)]² dx

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

1. Disk Method: Solid disk, R(x) = radius from axis to curve

2. Washer Method: Hollow center, R(x) = outer radius, r(x) = inner radius

3. Choose Integration Variable: Integrate with respect to variable that makes R(x) simpler

Volumes w/ Cylindrical Shells

V = 2π ∫[a to b] r(x) · h(x) dx

1. r(x) = distance from axis of rotation to shell

2. h(x) = height of shell = |f(x) - g(x)|

3. Use When: Disk/washer method gives complicated setup

Arc Length

L = ∫[a to b] √(1 + (dy/dx)²) dx

L = ∫[c to d] √((dx/dy)² + 1) dy

1. Choose the Form: That makes dy/dx or dx/dy simpler

2. Parametric Form: L = ∫√((dx/dt)² + (dy/dt)²) dt

3. Common Mistake: Don't forget the square root covers the entire expression

Applications to Physics & Engineering

Work = Force × Distance Applications:

1. Variable Force: W = ∫F(x) dx where F(x) changes with position

Spring work: W = ∫₀ᵈ kx dx = ½kd² (Hooke's Law: F = kx)

Pumping liquid: W = ∫ρg·h(x)·A(x) dx

2. Center of Mass: x̄ = (∫x·ρ(x) dx)/(∫ρ(x) dx)

3. Fluid Pressure: F = ∫ρg·h(y)·w(y) dy

ρ = density, g = gravity, h(y) = depth, w(y) = width at depth y

Sequences

1. Convergence: lim[n→∞] aₙ = L (finite number) → sequence converges to L

2. Divergence: lim[n→∞] aₙ = ∞, -∞, or doesn't exist → sequence diverges

3. Squeeze Theorem: If aₙ ≤ bₙ ≤ cₙ and lim aₙ = lim cₙ = L, then lim bₙ = L

Series

Convergence vs Divergence Explained:

Series CONVERGES: Partial sums Sₙ approach a finite limit

Series DIVERGES: Partial sums grow without bound or oscillate

If Σaₙ converges, then aₙ → 0 (but aₙ → 0 doesn't guarantee convergence!)

1. Geometric Series: Σar^n converges when |r| < 1, sum = a/(1-r)

2. p-Series: Σ(1/n^p) converges when p > 1, diverges when p ≤ 1

3. nth Term Test (Divergence Test):

If lim[n→∞] aₙ ≠ 0, then Σaₙ DIVERGES

If lim[n→∞] aₙ = 0, test is INCONCLUSIVE (might converge or diverge)

This test can ONLY prove divergence, never convergence

Other Convergence Tests

1. Ratio Test: lim[n→∞] |aₙ₊₁/aₙ| = L

L < 1: converges, L > 1: diverges, L = 1: inconclusive

Best for series with factorials or exponentials

2. Integral Test: ∫₁^∞ f(x) dx and Σf(n) both converge or both diverge

Use when f(x) is positive, decreasing, and continuous

3. Comparison Tests:

Direct: If 0 ≤ aₙ ≤ bₙ and Σbₙ converges, then Σaₙ converges

Limit: lim[n→∞] aₙ/bₙ = L > 0, then both series have same behavior

Power Series

Σcₙ(x - a)^n = c₀ + c₁(x - a) + c₂(x - a)² + ...

1. Radius of Convergence R: Use ratio test on coefficients

R = lim[n→∞] |cₙ/cₙ₊₁| or R = 1/lim[n→∞] |cₙ₊₁/cₙ|

2. Interval of Convergence: (a - R, a + R), then check endpoints separately

3. Operations: Can add, multiply, differentiate, integrate term by term within radius

Functions as Power Series

1. Geometric Series Pattern: 1/(1-x) = Σx^n for |x| < 1

2. Term-by-Term Operations: d/dx[Σcₙx^n] = Σncₙx^(n-1)

3. Integration: ∫[Σcₙx^n]dx = Σ[cₙx^(n+1)/(n+1)] + C

Taylor and McLaurin Series

f(x) = Σ[f^(n)(a)/n!](x - a)^n

Essential McLaurin Series (a = 0):

e^x = Σx^n/n! = 1 + x + x²/2! + x³/3! + ...

sin x = Σ(-1)ⁿx^(2n+1)/(2n+1)! = x - x³/3! + x⁵/5! - ...

cos x = Σ(-1)ⁿx^(2n)/(2n)! = 1 - x²/2! + x⁴/4! - ...

1/(1-x) = Σx^n = 1 + x + x² + x³ + ... for |x| < 1

ln(1+x) = Σ(-1)^(n+1)x^n/n = x - x²/2 + x³/3 - ... for |x| < 1

1. Taylor Centered at a: Use derivatives evaluated at x = a

2. Error Estimation: |Rₙ(x)| ≤ M|x - a|^(n+1)/(n+1)! where |f^(n+1)(c)| ≤ M

Parametric Curves

x = f(t), y = g(t)

1. Think of t as Time: Particle traces path as t increases

2. Eliminate Parameter: Solve for t in one equation, substitute into other

3. Direction of Motion: dx/dt and dy/dt tell you horizontal and vertical velocity

Calculus with Parametric Curves

dy/dx = (dy/dt)/(dx/dt)

d²y/dx² = d/dx[dy/dx] = [(d²y/dt²)(dx/dt) - (dy/dt)(d²x/dt²)]/(dx/dt)³

1. First Derivative: Chain rule applied to parametric equations

2. Second Derivative: More complex, use quotient rule carefully

3. Arc Length: L = ∫√((dx/dt)² + (dy/dt)²) dt

Polar Coordinates

x = r cos θ, y = r sin θ

r² = x² + y², tan θ = y/x

1. r = distance from origin, θ = angle from positive x-axis

2. Common Polar Curves:

Circle: r = a

Cardioid: r = a(1 ± cos θ) or r = a(1 ± sin θ)

Rose: r = a cos(nθ) or r = a sin(nθ)

3. Symmetry Tests: Replace (r,θ) with (-r,θ+π), (r,-θ), (-r,-θ)

Areas and Length in Polar Coordinates

Area = (1/2)∫[α to β] r² dθ

Arc Length = ∫[α to β] √(r² + (dr/dθ)²) dθ

1. Area Between Curves: (1/2)∫[r₁² - r₂²] dθ where r₁ ≥ r₂

2. Finding Intersection Points: Set r₁(θ) = r₂(θ) and solve for θ

3. Don't Forget the (1/2): This factor comes from the area of a sector

CALC 2 - COMPLETE SURVIVAL GUIDE