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Techniques of Integration and Applications in Calculus

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Termini in questo insieme (20)

  • What is the LIATE rule used for in integration by parts?
    LIATE helps choose u in integration by parts: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential, in that order of priority.
  • How do you complete the square for a quadratic expression ax^2 + bx + c?
    Rewrite as a(x + b/(2a))^2 + (c - b^2/(4a)) to simplify integration involving quadratics.
  • What substitution is used in trigonometric substitution for integrals involving sqrt(a^2 - x^2)?
    Use x = a sin θ, with θ in [-π/2, π/2], to simplify the integral.
  • State the product-to-sum identity for ∫ sin(mx) cos(nx) dx.
    sin A cos B = 1/2 [sin(A - B) + sin(A + B)].
  • What is the formula for integration by parts?
    ∫ u dv = uv - ∫ v du.
  • How do you integrate powers of sine and cosine when one power is odd?
    Express the odd power as (sin^2 x)^k sin x or (cos^2 x)^k cos x, use Pythagorean identity, and substitute u = cos x or sin x.
  • What is the trapezoidal rule formula for approximating ∫_a^b f(x) dx?
    T = (Δx/2) [f(x_0) + 2f(x_1) + 2f(x_2) + ... + 2f(x_{n-1}) + f(x_n)].
  • How is Simpson's rule expressed for approximating ∫_a^b f(x) dx?
    S = (Δx/3) [f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + ... + 4f(x_{n-1}) + f(x_n)], with n even.
  • What is the error bound for the trapezoidal rule?
    |E_T| ≤ (M(b - a)^3) / (12 n^2), where M bounds |f''(x)| on [a,b].
  • What is the error bound for Simpson's rule?
    |E_S| ≤ (M(b - a)^5) / (180 n^4), where M bounds |f^{(4)}(x)| on [a,b].
  • How do you determine if an improper integral of Type I converges?
    Evaluate the limit of the integral as the bound approaches infinity; if the limit exists and is finite, it converges.
  • What is the difference between Type I and Type II improper integrals?
    Type I has infinite limits of integration; Type II has discontinuities (vertical asymptotes) within the interval.
  • How do you handle an improper integral with a vertical asymptote inside the interval?
    Split the integral at the discontinuity and take limits approaching the asymptote from each side.
  • What is the general form for partial fraction decomposition with distinct linear factors?
    Express as sum of A_i / (a_i x + b_i) for each linear factor.
  • How do you decompose a rational function with repeated linear factors?
    Include terms for each power up to the multiplicity: A_1/(x-r) + A_2/(x-r)^2 + ... + A_r/(x-r)^r.
  • How do you decompose a rational function with irreducible quadratic factors?
    Use terms of the form (Ax + B) / (quadratic factor).
  • What is the substitution for integrals involving sqrt(x^2 - a^2)?
    Use x = a sec θ.
  • What is the substitution for integrals involving sqrt(a^2 + x^2)?
    Use x = a tan θ.
  • How do you find the average value of a function f on [a,b]?
    Average value = (1/(b - a)) ∫_a^b f(x) dx.
  • What is the formula for the volume of a solid of revolution about the y-axis using the shell method?
    V = 2π ∫_a^b (radius)(height) dx.