Calculus II Midterm #2 Review

• §7.7: Approximating Integrals
  • Any integral can be thought of as area under a curve.
  • Area can be approximated by rectangles, trapezoids, or parabolas.
  • Midpoint Rule: Δx [f(x_{mid 1})+f(x_{mid 2})+...+f(x_{mid n})]
  • Trapezoid Rule: Δx/2 [f(x₀)+2f(x₁)+2f(x₂)+...+2f(x_n-1)+f(x_n)]
  • Simpson's Rule: Δx/3 [f(x₀)+4f(x₁)+2f(x₂)+4f(x₃)+...+4f(x_n-1)+f(x_n)]


• §7.8: Improper Integrals
  • An integral is "improper" if interval begins or ends at ±∞
  • or includes an x-value for which the function is undefined.
  • Replace the problematic value with t and take the limit.
  • Result may be convergent (to a specific finite value) or divergent.

• §8.1: Arc Length
  • S = ∫ ds; the trick is finding the right form of ds!
  • In general: ds² = dx² + dy², so in Cartesian coordinates:
  • ds = √( 1 + (dy/dx)² ) dx     or     ds = √( 1 + (dx/dy)² ) dy


• §8.2: Area of a Surface of Revolution
  • Slice surface into narrow bands, and treat each as frustrum of a cone.
  • Each band has approx. area 2π r Δs (r = radius from axis to curve),
  • so total area is ∫ 2π r ds, with ds as defined above.


• §8.3: Moment and Center of Mass
  • Understand physical meaning of "moment" and "center of mass" (or "centroid").
  • Moments about axes:   M_x = ρ ∫_a^b ½ f(x)² dx     and     M_y = ρ ∫_a^b x f(x) dx
  • Center of mass occurs at (x,y), where x = M_y / mass     and     y = M_x / mass
  • Note that mass = ρ · area = ρ · ∫_a^b f(x) dx, so the ρ cancels anyway.

• §10.1: Parametric Equations
  • Instead of defining y as a function of x, we define x and y as functions of t.
  • Very useful in physics: t often represents time (but doesn't have to).
  • Sometimes y can be written as a function of x if x(t) is invertible.


• §10.2: Calculus with Parametric Curves
  • First Derivative: dy/dx = (dy/dt) / (dx/dt) ...because of chain rule.
  • Second Derivative: d²y/dx² = (d (dy/dx) / dt) / (dx/dt)
  • (Note that the derivatives are functions of t, not of x or y.)
  • Area: A = ∫_α^β y(t) x'(t) dt ...because dx = x'(t) dt (chain rule again)
  • Path Length: ∫ ds as usual, but ds = √( (dx/dt)² + (dy/dt)² ) dt.
  • (Note that the limits of integration here are t-values, not x-values.)


• §10.3: Polar Coordinates
  • Each point described as (r, θ); r = radius from origin; θ = angle
  • Caution: names are not unique. (r, θ+2π) is equivalent to (r, θ)
  • and (r, θ+π) is equivalent to (-r, &theta). Consider why geometrically.
  • Graphs: usually r is given as a function of θ. Make a data table and plot points.
  • Conversions: x = r cosθ, y = r sinθ, r² = x² + y², tanθ = y/x
  • Derivatives: dy/dx = (dy/dθ)/(dx/dθ) = (r'sinθ + rcosθ) / (r'cosθ - rsinθ)


• §10.4: Calculus with Polar Curves
  • Area: Slice region into wedges and treat each wedge as a sector of a circle.
  • Sector area is ½ r² Δθ, so total area is ∫_α^β ½ r² dθ.
  • Arc Length: as always, ∫ ds. This time ds = √( r² + (dr/dθ)² ) dθ
  • because everything else cancels out in an awesome way (try it!).

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