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Area between curves

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lesson from AP Calc

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Area between curves

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

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