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4010-Properties of the Definite Integral (5.3)

4010-Properties of the Definite Integral (5.3). BC Calculus. Properties of Definite Integrals. Think rectangles Distance. A) B) C) D). f (x). a dx b. Properties of Definite Integrals. Think rectangles Distance. a c b. E) NOTE : Same Interval.

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4010-Properties of the Definite Integral (5.3)

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  1. 4010-Properties of the Definite Integral (5.3) BC Calculus

  2. Propertiesof Definite Integrals • Think rectangles • Distance A) B) C) D) f (x) adxb

  3. Propertiesof Definite Integrals • Think rectangles • Distance acb E) NOTE: Same Interval (1). Shows the method to work Definite Integrals – like Σ (2). IMPORTANT: Finding Area between curves.

  4. Propertiesof Definite Integrals • Think rectangles • Distance acb F) If c is between a and b , then: Placement of c important: upper bound of 1st, lower bound of 2nd. REM: The Definite Integral is a number, so may solve the above like an equation.

  5. Examples: Show all the steps to integrate.

  6. Examples: GIVEN: 1) 2) 3)

  7. Examples: (cont.) GIVEN: 4) 5)

  8. Propertiesof Definite Integrals * Think rectangles Distance acb G) Iff(min)is the minimum value off(x)andf(max) is themaximum value of f(x) on the closed interval [a,b], then

  9. Example: Show that the integral cannot possibly equal 2. Show that the value of lies between 2 and 3

  10. AVERAGE VALUE THEOREM (for Integrals) Remember the Mean Value Theorem for Derivatives. And the Fundamental Theorem of Calculus Then:

  11. AVERAGE VALUE THEOREM (for Integrals) f (c) f (c)is the average of the function under consideration i.e. On the velocity graph f (c)is the average velocity (value). cis where that average occurs.

  12. AVERAGE VALUE THEOREM (for Integrals) f (c) f (c)is the average of the function under consideration NOTICE: f (c) is the height of a rectangle with the exact area of the region under the curve.

  13. Method: Find the average value of the function on [ 2,4].

  14. Example 2: A car accelerates for three seconds. Its velocity in meters per second is modeled by on t =[ 1, 4]. Find the average velocity.

  15. Last Update: • 01/27/11 • Assignment: Worksheet

  16. At different altitudes in the earth’s atmosphere, sound travels at different speeds The speed of sound s(x) (in meters per second) can be modeled by: Example 3 (AP): Where x is the altitude in kilometers. Find the average speed of sound over the interval [ 0,80 ]. SHOW ALL PROPERTY STEPS .

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