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Area Between a Continuous Function and x-Axis

Area Between a Continuous Function and x-Axis. Trip from CC-San Antonio. Make a narrative for the trip/Average velocity. Units of Line and Units of Area . Rise/run = miles/hour. Fill region with rectangles to estimate area . Units of area = length*width = hours*miles .

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Area Between a Continuous Function and x-Axis

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  1. Area Between a Continuous Function and x-Axis

  2. Trip from CC-San Antonio Make a narrative for the trip/Average velocity

  3. Units of Line and Units of Area Rise/run = miles/hour Fill region with rectangles to estimate area Units of area = length*width = hours*miles

  4. Questions to discuss • What is the average velocity (rate of change of distance) for the whole trip? • What is the average distance from the car to CC during the traveled time? • What degree polynomial would you use to fit this graph?

  5. Goal Find the area between the graph and the x-axis

  6. Nice Examples where Area is needed • Bicycle 1 Do rides 1, 3, 5 • Bicycle 2

  7. Area vs. Absolute Area Area of a rectangle = length*width or base*height Base 5, height 1 Area = 5 Absolute Area =5 Base 5, height -2 Area = -10 Absolute area= 10

  8. Estimating Areas • Grid (boxes) • Rectangles • Trapezoids - Boxes- • Indivisibles

  9. An Example to Understand the Techniques AREA BETWEEN A STRAIGHT LINE AND THE X-AXIS ON A CLOSED INTERVAL

  10. Use basic geometry to calculate this area

  11. Over/Underestimate Maximum Height Minimum Height Minimum height*Width<Area<Maximum Height*Width

  12. Grid (Boxes) Make a grid. Estimate the number of rectangles needed to fill up the region Estimate area of one rectangle Area ≅ No Boxes*Area one box

  13. Rectangles (Right/Left) Height at left hand points Height at right hand points

  14. Left-Hand Sums Write the summation with 20 subdivisions and use calculator to find the sum.

  15. Right-Hand Sums Write the summation with 20 subdivisions and use calculator to find the sum.

  16. Trapezoid area of a trapezoid with heights A, B, and width C is given by (A+B)/2*C

  17. Indivisibles Indivisible at each end point Average of all Indivisibles The area of the rectangle obtained above is the Average value of the heights multiplied by the length of the interval.

  18. The area between the graph of the continuous function y=f(x)and the x-axis on the interval [a ,b] is denoted Intuition about the geometry of integrals

  19. Indivisibles and Average Value of a function

  20. Exercise 2 Find an overestimate and underestimate for the 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.

  21. 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).

  22. Exercise 4

  23. 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.

  24. Area Application #1

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