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Unit E Overview

This overview explores the application of integration in modeling the effects of acceleration in a car, including calculating the car's speed and distance traveled. It also covers other integral applications such as consumption over time, area between curves, and volumes of solids.

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Unit E Overview

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  1. Unit E Overview

  2. Application of Integral

  3. Modeling the Effects of Acceleration • A car moving with initial velocity of 5 mph accelerates at the rate of a(t) = 2.4t mph per second for 8 seconds. How fast is the car going when the 8 seconds are up? • If the initial velocity is 5 mph then 5 + 76.8 = 81.8 mph

  4. Anytime you want to know how far, how much or how many, you use an integral.

  5. Modeling the Effects of Acceleration • A car moving with initial velocity of 5 mph accelerates at the rate of a(t) = 2.4t mph per second for 8 seconds. How far did the car travel in those 8 seconds? distance = 244.8 mph X seconds Convert to miles by multiplying by 1/3600 = .068 miles

  6. Consumption over time • From 1970 to 1980, the rate of potato consumption in a particular country was C(t) = 2.2 + 1.1t millions of bushels per year. • How many millions of bushels were produced from the beginning of 1972 to the beginning of 1974? • Solution: we seek the cumulative effective of the consumption rate for 2 < t < 4.

  7. Example a

  8. Example d

  9. Example a

  10. Example c

  11. Example – no calc d

  12. Daily: A honey bee makes several trips from the hive to a flower garden. The velocity graph is shown below. What is the total distance traveled by the bee? What is the displacement of the bee? What is the position of the bee?

  13. HW • Math Pong Worksheet

  14. Applications of Definite Integrals

  15. Area Between Curves • If f and g are continuous with f(x) > g(x) through [a,b], then the area between the curves y = f(x) and y = g(x) from a to b is the integral of [f-g] from a to b,

  16. Area of an Enclosed Region • When a region is enclosed by intersecting curves, the intersection points give the limits of integration. • Find the area of the region enclosed by the parabola y = 2 – x2 and the line y = -x. • Solution: graph the curves to determine the x-values of the intersections and if the parabola or the line is on top.

  17. Using a Calculator • Find the area of the region enclosed by the graphs of y = 2cosx and y = x2-1. • Solution: Graph to determine the intersections. • X = ± 1.27 • 4.99 units2

  18. Boundaries with Changing Functions Region A: • If a boundary of a region is defined by more than one function, we can partition the region into subregions that correspond to the function changes. • Find the area of the region R in the first quadrant bounded by y = (x)1/2 and below by the x-axis AND the line y = x -2 Region B: B A A +B = 3.33

  19. HW • Pg. 452 # 3, 5, 7, 14 (a) , 15, 19, 35

  20. Volumes

  21. Volume of a Solid • The volume of a solid of a known integrable cross section area A(x) from x = a to x = b is the integral of A from a to b:

  22. How to Find the Volume by the Method of Slicing • Sketch the solid and a typical cross section. • Find a formula for A(x). • Find the limits of integration. • Integrate A(x) to find the volume.

  23. Known Cross Sections • Knows Cross Sections Play doh paper activity

  24. Solid of Revolution • Unknown cross sections, you must figure out.

  25. Circular Cross SectionsEx: A Solid of Revolution • The region between the graph f(x) = 2 + xcosx and the x-axis over the interval [-2, 2] is revolved about the x-axis to generate a solid. Find the volume of the solid. (Note x-scale is .5)

  26. If you were to revolve the figure, the cross sections would would be circular. • The area of a circle is πr2 . • The radius of each circle will be the equation that has been given: 2 + xcosx(Note, the given function will ALWAYS be the radius when revolving an equation around the x-axis.) • A(x) = π(2+xcosx)2 • Set up a definite integral • Evaluate using the calculator: • The volume is 52.43 units3

  27. Example Circular Cross Section Determine the volume of the solid obtained by rotating the region bounded by y = x² - 4x + 5, x = 1, x = 4, and the x-axis, about the x axis.

  28. Drill • The region bounded by the curve y = x2 + 1 and the line y = -x + 3 is revolved about the x-axis to generate a solid. Find the volume of the solid. (See the “washer” example from notes.) 23.4π units3

  29. HW • Pg. 463 # 1, 3, 5

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