Calc 2 flash

Notation: ∫f(x)dx represents?

The general antiderivative of f(x), denoted F(x) + C.

Notation: ∫[a,b] f(x)dx represents?

The exact signed area between f(x) and the x-axis on the interval [a, b].

Property: ∫[a,b] k⋅f(x) dx = ?

k ⋅ ∫[a,b] f(x) dx

Property: ∫[a,b] (f(x) + g(x)) dx = ?

∫[a,b] f(x) dx + ∫[a,b] g(x) dx

Property: ∫[a,a] f(x) dx = ?

0

Property: ∫[a,c] f(x) dx + ∫[c,b] f(x) dx = ?

∫[a,b] f(x) dx

Property: ∫[b,a] f(x) dx = ?

- ∫[a,b] f(x) dx

Fundamental Theorem of Calculus (Part 1 - Derivative Form): d/dx [ ∫[a,x] f(t) dt ] = ?

f(x)

Fundamental Theorem of Calculus (Part 2 - Evaluation Form): How to evaluate ∫[a,b] f(x) dx if F'(x) = f(x)?

F(b) - F(a)

FTC with Chain Rule: d/dx [ ∫[a, g(x)] f(t) dt ] = ?

f(g(x)) ⋅ g'(x)

Area Between Curves (Functions of x): Area between f(x) (top) and g(x) (bottom) from a to b?

∫[a,b] (f(x) - g(x)) dx

Area Between Curves (Functions of y): Area between f(y) (right) and g(y) (left) from c to d?

∫[c,d] (f(y) - g(y)) dy

Volume: Disk Method (Rotate about x-axis): Rotate f(x) about x-axis on [a,b]. Volume = ?

∫[a,b] π [f(x)]² dx

Volume: Washer Method (Rotate about x-axis): Rotate region between R(x) (outer radius) and r(x) (inner radius) about x-axis on [a,b]. Volume = ?

∫[a,b] π ([R(x)]² - [r(x)]²) dx (Remember R, r depend on axis of rotation)

Volume: Disk/Washer Method (Rotate about y-axis): General form?

Integrate with dy. Volume = ∫[c,d] π [R(y)]² dy or ∫[c,d] π ([R(y)]² - [r(y)]²) dy

Volume: Shell Method (Rotate about y-axis): Rotate f(x) (height) at distance x (radius) from y-axis on [a,b]. Volume = ?

∫[a,b] 2π x f(x) dx (Adjust radius x and height f(x) for axis x=k or regions between curves)

Volume: Shell Method (Rotate about x-axis): General form?

Integrate with dy. Volume = ∫[c,d] 2π y f(y) dy (Adjust radius y and height f(y) for axis y=k or regions between curves)

Work done by variable force F(x) from a to b?

Work = ∫[a,b] F(x) dx

Average Value of f(x) on [a, b] (f_avg)?

f_avg = (1 / (b-a)) * ∫[a,b] f(x) dx

Mean Value Theorem (MVT) for Integrals guarantees...?

There exists a c in [a,b] such that f(c) = f_avg.

Integration by Parts Formula: ∫ u dv = ?

uv - ∫ v du

IBP: Strategy for choosing u? (Acronym)

LIPET: Logarithmic, Inverse Trig, Polynomial, Exponential, Trig (Choose first type present as u)

Trig Identity: sin²x + cos²x = ?

1 (Know rearrangements: sin²x = 1-cos²x, cos²x = 1-sin²x)

Trig Identity: 1 + tan²x = ?

sec²x (Know rearrangement: tan²x = sec²x - 1)

Trig Half-Angle: sin²x = ?

(1 - cos(2x)) / 2

Trig Half-Angle: cos²x = ?

(1 + cos(2x)) / 2

Trig Integral Strategy: ∫ sin^m x cos^n x dx, one power odd?

Save one odd factor, convert rest using sin²x+cos²x=1, let u = other trig function.

Trig Integral Strategy: ∫ sin^m x cos^n x dx, both powers even?

Use half-angle formulas repeatedly.

Trig Integral Strategy: ∫ tan^m x sec^n x dx, n (secant power) even & ≥2?

Save sec²x, convert rest sec²x to 1+tan²x, let u = tan x.

Trig Integral Strategy: ∫ tan^m x sec^n x dx, m (tangent power) odd & ≥1? (Need ≥1 secant)

Save sec x tan x, convert rest tan²x to sec²x-1, let u = sec x.

Trig Sub: Integrand has sqrt(a² - x²)? Use x = ?

x = a sin θ

Trig Sub: Integrand has sqrt(a² + x²)? Use x = ?

x = a tan θ

Trig Sub: Integrand has sqrt(x² - a²)? Use x = ?

x = a sec θ

PFD Setup: Term for Linear Factor (ax+b)?

A / (ax+b)

PFD Setup: Terms for Repeated Linear Factor (ax+b)^k?

A₁/(ax+b) + A₂/(ax+b)² + ... + A_k/(ax+b)^k

PFD Setup: Term for Irreducible Quadratic (ax²+bx+c)?

(Ax+B) / (ax²+bx+c)

PFD Setup: Terms for Repeated Irred. Quad. (ax²+bx+c)^k?

(A₁x+B₁)/(ax²+bx+c) + ... + (A_kx+B_k)/(ax²+bx+c)^k

Integral of PFD term ∫ A/(mx+b) dx = ?

(A/m) ln|mx+b| + C

Integral of PFD term ∫ 1/(x²+a²) dx = ? (Arctan form)

(1/a) arctan(x/a) + C

Integral of PFD term ∫ x/(x²+a²) dx = ? (Log form)

(1/2) ln(x²+a²) + C

Improper Integral Evaluation ∫[a, ∞) f(x) dx = ?

lim (t→∞) ∫[a, t] f(x) dx

Improper Integral Evaluation ∫[a, b] f(x) dx (discontinuity at b) = ?

lim (t→b⁻) ∫[a, t] f(x) dx (Similar for discontinuity at a or c in between)

Arc Length Formula for y=f(x) on [a,b]? L = ?

L = ∫[a,b] sqrt(1 + [f'(x)]²) dx

Centroid coordinate x̄ (x-bar) = ?

x̄ = M_y / m = (Moment about y-axis) / (Total Mass/Area) x̄ = (∫ x [top(x)-bot(x)] dx) / (∫ [top(x)-bot(x)] dx)

Centroid coordinate ȳ (y-bar) = ?

ȳ = M_x / m = (Moment about x-axis) / (Total Mass/Area) ȳ = (∫ (1/2) ([top(x)]² - [bot(x)]²) dx) / (∫ [top(x)-bot(x)] dx)

Taylor Polynomial of degree d for f(x) centered at x=a, P_d(x) = ? (Sigma Notation)

P_d(x) = ∑[n=0 to d] (f^(n)(a) / n!) * (x-a)^n

Taylor Polynomial P_1(x) is...?

The tangent line to f(x) at x=a.

Sequence a_n converges if...?

lim (n→∞) a_n exists and is finite.

Series ∑ a_n converges if...?

The sequence of partial sums S_n = ∑[i=1 to n] a_i converges.

Geometric Series ∑[n=0 to ∞] ar^n converges when?

|r| < 1

Geometric Series ∑[n=0 to ∞] ar^n Sum (if convergent)?

S = a / (1-r) (where a is the first term)

Nth Term Test for Divergence?

If lim (n→∞) a_n ≠ 0, then ∑ a_n DIVERGES. (If limit IS 0, test is INCONCLUSIVE).

Integral Test Conditions?

Need f(x) corresponding to a_n to be: 1. Continuous, 2. Positive, 3. Decreasing (for x ≥ N).

Integral Test Conclusion?

∑ a_n and ∫[N, ∞) f(x) dx either both converge or both diverge.

p-Series ∑ 1/n^p converges when? Diverges when?

Converges if p > 1. Diverges if p ≤ 1.

Direct Comparison Test (DCT)?

If 0 ≤ a_n ≤ b_n: ∑ b_n converges ⇒ ∑ a_n converges. If 0 ≤ b_n ≤ a_n: ∑ b_n diverges ⇒ ∑ a_n diverges.

Limit Comparison Test (LCT)?

Given a_n > 0, b_n > 0. Find L = lim (n→∞) (a_n / b_n). If 0 < L < ∞, then ∑ a_n and ∑ b_n share the same fate (both converge or both diverge).

Alternating Series Test (AST) Conditions for ∑ (-1)^n b_n (b_n > 0)?

Converges if: 1. b_{n+1} ≤ b_n (decreasing) eventually. 2. lim (n→∞) b_n = 0.

Define: Absolute Convergence for ∑ a_n.

∑ |a_n| converges.

Define: Conditional Convergence for ∑ a_n.

∑ a_n converges, BUT ∑ |a_n| diverges.

Ratio Test? (Use for factorials, k^n)

Find L = lim (n→∞) |a_{n+1} / a_n|. L < 1 ⇒ Abs. Conv. | L > 1 ⇒ Div. | L = 1 ⇒ Inconclusive.

Root Test? (Use for terms like (expr)^n)

Find L = lim (n→∞) |a_n|^(1/n). L < 1 ⇒ Abs. Conv. | L > 1 ⇒ Div. | L = 1 ⇒ Inconclusive.

General form of Power Series centered at a?

∑[n=0 to ∞] c_n (x-a)^n

How to find Radius of Convergence (R) for a Power Series?

Usually use the Ratio Test (or Root Test) on ∑ |c_n (x-a)^n|, solve L < 1 for |x-a|.

How to find Interval of Convergence (IOC) for a Power Series?

1. Find Radius R. 2. Interval is initially (a-R, a+R). 3. Test convergence at the endpoints x = a-R and x = a+R using other series tests (p-series, AST, etc.). Include endpoints if they converge.