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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 Presentation
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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
• 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?
GoalFind the area between the graph and the x-axis
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
Estimating Areas
• Grid (boxes)• Rectangles• Trapezoids - Boxes-• Indivisibles
An Example to Understand the Techniques
AREA BETWEEN A STRAIGHT LINE AND THE X-AXIS ON A CLOSED INTERVAL
Use basic geometry to calculate this area
Over/Underestimate
Minimum Height
Maximum Height
Minimum height*Width<Area<Maximum Height*Width
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≅
Rectangles (Right/Left)
Height at left hand points Height at right hand points
Left-Hand Sums
Write the summation with 20 subdivisions and use calculator to find the sum.
Right-Hand Sums
Write the summation with 20 subdivisions and use calculator to find the sum.
Trapezoid
area of a trapezoid with heights A, B, and width C is given by (A+B)/2*C
Indivisibles
Indivisible at each end point Average of all Indivisibles
The area of the rectangle obtained above is approximated by calculating the Average value of the heights at the end
points and multiply the average by the length of the interval.
The area between the graph of the continuous function y=f(x)and the x-axis on the interval [a ,b] is denoted
Indivisibles and Average Value of a function
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.
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).
Exercise 4
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.
Area Application #1