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Area between curves
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Area between curves

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lesson from AP Calc

lesson from AP Calc

Published in: Sports, Technology

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  • 1. Area between Curves
  • 2. How do we define area “between” curves?
    • Two non-intersecting functions with arbitrary boundaries
    • Two functions that have 2 or more points of intersection
    • “ Triangular” regions bounded by three functions
  • 3. Two non-intersecting functions with arbitrary boundaries
    • Ex 1: Find the area bounded by f(x) = x 2 +2, g(x) = -x, x = 0, and x = 1
    • Sketch the region(s)
    • Draw representative rectangle
    • Determine dx or dy
  • 4. Two functions that have 2 points of intersection
    • Ex 2- Find the area bounded by
    • f(x) = 2 – x 2 and g(x) = x
    • Sketch the region
    • Find the points of intersection
      • f(x) = g(x) usually works
    • Draw representative rectangle
    • Determine dx or dy
  • 5. Two functions that have more than 2 points of intersection
    • Ex 3- Find the area between the graphs of
    • f(x) = 3x 3 -x 2 -10x and
    • g(x) = -x 2 + 2x
    • Sketch the region(s)
    • Find the points of intersection
      • f(x) = g(x) usually works
    • Draw representative rectangle
    • Determine dx or dy
  • 6. When do we ever use dy?
    • Glad you asked 
    • Ex 4- Find the area bounded by
    • x = 3 – y 2 and y = x - 1
    • Sketch the region
    • Find the points of intersection
      • Solve a system of equations
    • Draw representative rectangle
    • Determine dx or dy
  • 7. “Triangular” regions bounded by three functions
    • Ex 5- Find the area of the region bounded by: y = x, y = 2 – x,
    • and y = -1
    • Sketch the region
    • Find the points of intersection
      • Solve a system of equations
    • Can this be done geometrically?
    • Draw representative rectangle
    • Determine dx or dy
    • Can this be done in terms of the other variable?
  • 8. Quiz next session (45 min. maximum)
    • Riemann Sums (left, right, midpoint, inscribed, circumscribed)
    • Trapezoidal Rule
    • Definite Integrals
    • Fundamental Theorem of Calculus (I and II)
    • Average Value of Function