# Lecture 8 - CC2053 Calculus 8 Introduction to Calculus and...

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1 CC2053: Calculus 8 Introduction to Calculus and Linear Algebra 2011/2012 Semester 1 INTEGRATION: Definite Integrals

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2 CC2053: Calculus 8 Definition of a Definite Integral Let f be a continuous defined on the closed interval a x b , and let 1. a = x 0 < x 1 < ····· < x n-1 < x n = b 2. Δ x k = x k x k -1 , for k = 1, 2, ··· , n 3. Δ x k 0 as n →∞ 4. for k = 1, 2, ··· , n
3 CC2053: Calculus 8 Definition of a Definite Integral Then is called definite integral of f from a to b. The integrand is f (x), the lower limit is a , and the upper limit is b.

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4 CC2053: Calculus 8 Area Under a Curve
5 CC2053: Calculus 8 Area Under a Curve If f ( x ) is continuous f ( x ) 0 on the interval a x b , then the region under the curve y = f ( x ) above a x b has area

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6 CC2053: Calculus 8 Fundamental Theorem of Calculus If f ( x ) is continuous function on the closed interval a x b and F ( x ) is any antiderivative of f ( x ) then
7 CC2053: Calculus 8 Example 9 Evaluate a. b. c.

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8 CC2053: Calculus 8 Example 9:- solution a. Let b. Then
9 CC2053: Calculus 8 Example 9:- solution c.

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10 CC2053: Calculus 8 Area between a Curve and the x -axis For a function f continuous over [ a , b ], the area between y = f ( x ) and the x axis from x = a to x = b can be found using definite integrals as follows: For over [ a , b ] For over [ a , b ]
11 CC2053: Calculus 8 Example 10 Find the area bounded by and for .

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12 CC2053: Calculus 8 Example 10:- solution Area -2 2 y = x 2 -4 x y
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Lecture 8 - CC2053 Calculus 8 Introduction to Calculus and...

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