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6036: Area of a Plane Region AB Calculus

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

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Area 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 Remember the 7 step method. b = Perpendicular to the axis! h = Height is always Top minus Bottom! a b Area of rectangle

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

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

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Area between curves REPEAT: Height is always Top minus Bottom! a b f (x) g (x) Height of rectangle

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

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

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

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

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

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Multiple Regions 1)Find the points of intersections to determine the intervals. 2)Find the heights (Top minus Bottom) for each region. 3)Use the Area Addition Property. a b c b = h = f (x) g (x)

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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!

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

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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

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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).

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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 ]

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

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