Calculus II Midterm #1 Review

• § 6.1: Area between Curves
  • Vertical rectangles: ∫_a^b (top curve – bottom curve) dx
  • Horizontal rectangles: ∫_a^b (right curve – left curve) dy
  • Be careful about piecewise functions and change-overs between what's on top (or right)


• §6.2: Solids of Rotation using Discs/Washers
  • Rectangles perpendicular to axis of rotation produce discs or washers
  • Volume of each washer is π·(r²_out – r²_in)·Δx (...or Δy)
  • Be careful with radii; they should be distance between axis of rotation and each curve
  • Total volume should be π·∫_a^b(r²_out – r²_in) dx (...or dy)


• §6.3: Solids of Rotation using Shells
  • Rectangles parallel to axis of rotation produce hollow cylindrical shells
  • Volume of each shell is 2πr·h·Δx (...or Δy)
  • r is distance from axis of rotation: usually x (or y) minus the axis
  • h is the height of a rectangle: usually f(x) or f(x)-g(x) (...or y)
  • Total volume should be 2π·∫_a^br·h dx (...or dy)


• §6.4: Work
  • Work = Force · Distance (W=Fd)
  • If force is not constant, try subdividing the distance into Δx-size pieces,
  • find the work done over each tiny distance, and integrate
  • If the object is not a single point, try subdividing it into Δx-size pieces,
  • find the work done on each tiny piece, and integrate

• §5.5: Substitution
  • Useful for dealing with nested functions
  • Choose innermost function as u
  • Works if u' appears somewhere in integrand
  • For definite integrals, be careful with limits


• §7.1: Integration by Parts
  • Useful for dealing with products of functions
  • ∫ u dv = u·v – ∫ v du
  • Choose u that is made simpler by differentiating
  • Choose v that is made simpler by integrating
  • If cyclic, try setting equal to original and solving


• §7.2: Trigonometric Integrals
  • Useful for dealing with trigonometric functions raised to powers
  • sin^mx cosⁿx:
    • If one has an odd power, split one off and use Pythagorean Theorem and u-sub.
    • If both even, use half-angle formulas
  • tan^mx secⁿx:
    • If sec has an even power, split sec²x and convert others to tan
    • If tan has odd power, split secx tanx and convert others to sec


• §7.3: Trigonometric Inverse Substitution
  • Useful for dealing with √(±x²±a²)
  • a²–x²: use x = a·sinθ
  • x²–a²: use x = a·secθ
  • x²+a²: use x = a·tanθ


• §7.4: Partial Fraction Decomposition
  • Useful for dealing with rational expressions
  • If improper, divide first with polynomial long division
  • Factor denominator and "split up" into simpler fractions
  • Split up integral and integrate each piece separately


• §7.5: Strategy for Integration
  • Know how to use each method and when each would be useful
  • Some integrals may require combining several methods together

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