Formulae Flashcards
What is the area under a curve y = f(x) from x = a to x = b?
A = ∫ₐᵇ f(x) dx
What is the area enclosed by a polar curve r = r(θ) from θ = α to θ = β?
A = ½ ∫ₐᵝ [r(θ)]² dθ
What is the area under a parametric curve x = x(t), y = y(t) from t = t₁ to t = t₂?
A = ∫ₜ₁ᵗ₂ y(t)·x′(t) dt
What is the length of a curve y = f(x) from x = a to x = b?
L = ∫ₐᵇ √[1 + (f′(x))²] dx
What is the length of a parametric curve x = x(t), y = y(t) for t ∈ [t₁, t₂]?
L = ∫ₜ₁ᵗ₂ √[(x′(t))² + (y′(t))²] dt
What is the length of a polar curve r = r(θ) from θ = α to θ = β?
L = ∫ₐᵝ √[r(θ)² + (r′(θ))²] dθ
How do you compute the volume of revolution by washers about the x‑axis for y = f(x), a ≤ x ≤ b?
V = π ∫ₐᵇ [f(x)]² dx
How do you compute the volume of revolution by washers between outer radius R(x) and inner radius r(x)?
V = π ∫ₐᵇ [R(x)² – r(x)²] dx
How do you compute the volume of revolution by cylindrical shells about the y‑axis for y = f(x), a ≤ x ≤ b?
V = 2π ∫ₐᵇ x·f(x) dx
How do you evaluate limₓ→c f(x)/g(x) when both numerator and denominator → 0 or ±∞?
limₓ→c f(x)/g(x) = limₓ→c f′(x)/g′(x), provided the latter limit exists.
What is the mass of a planar lamina between y = g(x) and y = f(x) on [a,b] with density ρ?
M = ρ ∫ₐᵇ [f(x) – g(x)] dx
What is the moment about the y‑axis for that lamina?
M_y = ρ ∫ₐᵇ x·[f(x) – g(x)] dx
What is the moment about the x‑axis for that lamina?
M_x = ρ ∫ₐᵇ ½·[f(x)² – g(x)²] dx
What is the second moment of area about the y‑axis (I_y) for that lamina?
I_y = ρ ∫ₐᵇ x²·[f(x) – g(x)] dx
What is the second moment of area about the x‑axis (I_x) for that lamina?
I_x = ρ ∫ₐᵇ ⅓·[f(x)³ – g(x)³] dx
What is the First Fundamental Theorem of Calculus for F(x) = ∫ₐˣ f(t) dt?
F′(x) = f(x)
What is the Second Fundamental Theorem of Calculus for ∫ₐᵇ f(x) dx?
∫ₐᵇ f(x) dx = F(b) – F(a), where F′(x) = f(x)
How do you differentiate F(x), where the upper bound, b is u(x)? ∫ₐᵇ f(t) dt?
F′(x) = f(u(x))·u′(x)
