4033-Properties of the Definite Integral (5.3) AB Calculus.

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Presentation transcript:

4033-Properties of the Definite Integral (5.3) AB Calculus

Properties of Definite Integrals A) B) C) D) Think rectangles Distance a dx b f (x) a to a nowhere rectangle Opposite direction Constant multiplier

Properties of Definite Integrals Think rectangles Distance a c b E) NOTE: Same Interval (2).IMPORTANT: Finding Area between curves. (1). Shows the method to work Definite Integrals – like Σ subtract

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

Examples: Show all the steps to integrate. Step 1: Break into parts FTC rectangle Remove constant multiplier

Examples: GIVEN: 1) 2) 3)

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

Properties of Definite Integrals Distance * Think rectangles G) If f(min) is the minimum value of f(x) and f(max) is the maximum value of f(x) on the closed interval [a,b], then a c b

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

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

a c b

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). c is where that average occurs.

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.

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

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.

Last Update: 01/27/11 Assignment: Worksheet

Example 3 (AP): 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: Where x is the altitude in kilometers. Find the average speed of sound over the interval [ 0,80 ]. SHOW ALL PROPERTY STEPS.