6036: Area of a Plane Region AB Calculus. Accumulation vs. Area Area is defined as positive. The base and the height must be positive. Accumulation can.

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Presentation transcript:

6036: Area of a Plane Region AB Calculus

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)

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

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

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

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

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

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

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

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!

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

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)

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!

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

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

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).

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 ]

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