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Theorem Cauchy-Inequality for IntegrationDefinite Integrals
Definition Let
In the ith subinterval choose Remark For equal width, i.e. divide we have Choose
OR
Example Evaluate
Example
Example Using a definite integral, evaluate (a) (b)
AL95II(a) Evaluate (b) By considering a suitable definite integral, evaluate AL83II-1 Evaluate (a) (b) (c)
Example*
Properties of Definite Integrals
P1 The value of the definite integral of a given function is a real number, depending on its lower and upper limits only, and is independent of the choice of the variable of integration, i.e.
P2
P3
P4 Let Example (a)
P5* Comparison of two integrals
Example hence
Example Prove that (a) (b)
ExampleIn Figure, By considering the area of Hence show that
P6 Rules of Integration If (a) (b)
P7* (a) (b) (c) (d) Proof(a)
Example Evaluate (a)
Exercise 7C 5. By proving that evaluate (a)
6 (a) Show that (b) Using (a), or otherwise, evaluate the following integrals: (i)
Remind
Graph of an odd functionGraph of an even function
P8 (i) If then (ii) If then Proof
ExampleEvaluate (a)
Example Prove that (a)
Definition Let
P9 If (a) (b) (c) (d) Proof
Theorem Cauchy-Inequality for Integration If Proof
Example Date: 2016-04-22; view: 1051
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