Definite Integrals and Antiderivatives

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Presentation transcript:

Definite Integrals and Antiderivatives 5.3 Definite Integrals and Antiderivatives

Quick Review

Quick Review Solutions

What you’ll learn about Properties of Definite Integrals Average Value of a Function Mean Value Theorem for Definite Integrals Connecting Differential and Integral Calculus Essential Questions What are the properties of definite integrals and how do they help us to understand how definite integrals are connected to Integral Calculus?

Rules for Definite Integrals Order of Integration: Zero: Constant Multiple: Sum and Difference: Additivity:

Rules for Definite Integrals Max-Min Inequality: If max f and min f are the maximum and minimum values of f on [a, b], then Domination:

Example Using the Rules for Definite Integrals Suppose Find if possible. Find if possible. Find if possible. Not enough information.

Average (Mean) Value Example Applying the Definition Use NINT If f is integrable on [a, b], its average (mean) value on [a, b] is Example Applying the Definition Use NINT Find the average value of on [0, 4].

The Mean Value Theorem for Definite Integrals If f is continuous on [a, b], then at some point c in [a, b],

The Derivative of an Integral This means that the integral is an antiderivative of f. Set x = a

Pg. 290, 5.3 #1-35 odd