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6036: Area of a Plane Region

6036: Area of a Plane Region. AB Calculus. Accumulation vs. Area. Accumulation can be positive, negative, and zero. Area is defined as positive . The base and the height must be positive. h = always Top minus Bottom (Right minus Left).

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6036: Area of a Plane Region

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  1. 6036: Area of a Plane Region AB Calculus

  2. Accumulation vs. Area Accumulation can be positive, negative, and zero. Area is defined as positive. The base and the height must be positive. h = always Top minus Bottom (Right minus Left)

  3. DEFN: If f is continuous and non-negative on [ a, b ], the region R, bounded by f and the x-axis on [ a,b ] is Area Area of rectangle b = Perpendicular to the axis! h = Height is always Top minus Bottom! ab Remember the 7 step method.

  4. Ex: Find the Area of the region bounded by the curve, and the x-axis bounded by [ 0,  ]

  5. Ex: Find the Area of the region bounded by the curve, and the x-axis bounded by [ -1, 1 ]

  6. Area between curves f (x) g (x) a b Height of rectangle REPEAT: Height is alwaysTop minus Bottom!

  7. Area between curves Both below h=(0-g)-(0-f) h=f-g One above one below h=(f-0)+(0-g) h=f-g Both above h=f-g a b The location of the functions does not affect the formula. <Always Top-bottom>

  8. Area : Method: Find the area bounded by the curves and on the interval x = -1 to x = 2

  9. Area : Example (x-axis): Find the area bounded by the curves and

  10. Area: Working with y-axis Area between two curves. When working with y-axis, height is always Right minus Left. a b h (y) Perpendicular to y-axis! k (y) The location of the functions does not affect the formula.

  11. Area : Example (y-axis): Find the area bounded by the curves and Perpendicular to y-axis

  12. Multiple Regions • Find the points of intersections to determine the intervals. • Find the heights (Top minus Bottom) for each region. • Use the Area Addition Property. f (x) g (x) a b c b = h = h =

  13. Area : Example (x-axis - two regions): Find the area bounded by the curve and the x-axis. NOTE: The region(s) must be fully enclosed!

  14. Area : Example ( two regions): Find the area bounded by the curve and . NOTE: The region(s) must be fully enclosed!

  15. Area : Example (Absolute Value): Find the area bounded by the curve and the x-axis on the interval x = -2 and x = 3 PROBLEM 21

  16. Velocity and Speed: Working with Absolute Value DEFN: Speed is the Absolute Value of Velocity. The Definite Integral of velocity is NET distance (DISPLACEMENT). The Definite Integral of Speed is TOTAL distance. (ODOMETER).

  17. Total Distance Traveled vs. Displacement The velocity of a particle on the x-axis is modeled by the function, . Find the Displacement and Total Distance Traveled of the particle on the interval, t  [ 0 , 6 ]

  18. Updated: • 01/29/12 • Text p 395 # 1 – 13 odd • P. 396 # 15- 33 odd

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