Calculus II Midterm #1 Review
Calculus II Midterm #1 Review
Applications of Integrals
- § 6.1: Area between Curves
- Vertical rectangles: ∫ab (top curve – bottom curve) dx
- Horizontal rectangles: ∫ab (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·(r2out – r2in)·Δx (...or Δy)
- Be careful with radii; they should be distance between axis of rotation and each curve
- Total volume should be π·∫ab(r2out – r2in) 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·∫abr·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
Advanced Integration Techniques
- §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
- sinmx cosnx:
- If one has an odd power, split one off and use Pythagorean Theorem and u-sub.
- If both even, use half-angle formulas
- tanmx 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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