area between a continuous function and x-axis
DESCRIPTION
Area Between a Continuous Function and x-Axis. Trip from CC-San Antonio. Make a narrative for the trip. Units of Line and Area. Rise/run = miles/hour. Fill region with rectangles. Units of area = length*width = hours*miles. - PowerPoint PPT PresentationTRANSCRIPT
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Area Between a Continuous Function and x-Axis
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Trip from CC-San Antonio
Make a narrative for the trip
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Units of Line and Area Rise/run = miles/hour
Fill region with rectangles
Units of area = length*width = hours*miles
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• What is the average velocity (rate of change of distance)?
• What is the average distance from CC during the traveled time?
• What degree polynomial would you use to fit this graph?
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GoalFind the area between the graph and the x-axis
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Area vs. Absolute AreaArea of a rectangle = length*width or base*height
Base 5, height 1Area = 5Absolute Area =5
Base 5, height -2Area = -10Absolute area= 10
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Estimating Areas
• Grid (boxes)• Rectangles• Trapezoids - Boxes-• Indivisibles
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An Example to Understand the Techniques
AREA BETWEEN A STRAIGHT LINE AND THE X-AXIS ON A CLOSED INTERVAL
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Use basic geometry to calculate this area
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Over/Underestimate
Minimum Height
Maximum Height
Minimum height*Width<Area<Maximum Height*Width
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Grid (Boxes)
Make a grid. Estimate the number of rectangles needed to fill up the region Estimate area of one rectangleArea No Boxes*Area one box≅
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Rectangles (Right/Left)
Height at left hand points Height at right hand points
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Left-Hand Sums
Write the summation with 20 subdivisions and use calculator to find the sum.
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Right-Hand Sums
Write the summation with 20 subdivisions and use calculator to find the sum.
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Trapezoid
area of a trapezoid with heights A, B, and width C is given by (A+B)/2*C
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Indivisibles
Indivisible at each end point Average of all IndivisiblesThe area of the rectangle obtained above is the Average
value of the heights multiplied by the length of the interval.
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The area between the graph of the continuous function y=f(x)and the x-axis on the interval [a ,b] is denoted
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Indivisibles and Average Value of a function
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Exercise 2
i) Find an overestimate and underestimate for the ii) Estimate the area using seven subdivisions
a. Grid technique. b. Left-hand rectangles. c. Indivisibles.
For questions (ii) (a-c), also write the expression using summation notation.
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Exercise 3
i) Find and overestimate and lower estimate for the area. ii) Estimate the area using rectangles and twelve subdivisions (use summation notation and the calculator).
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Exercise 4
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Find the area of the region and use it to determine the average distance between the car and CC for
the whole length of the trip.
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Area Application #1