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m23 definite integrals and area of plane region

# m23 definite integrals and area of plane region - The...

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The Definite Integral and Area of a Plane Region

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Consider the region R bounded by the graph of f , the lines x = a and x = b , and the x -axis. R Δ i x = x i – x i- 1 i th rectangle: f ( ε i ) Δ i x x 1 x 2 x 3 x 0 x i -1 x i x n x n- 1 Δ 1 x Δ 2 x Δ 3 x Δ n x Divide the interval [ a , b ] into n subintervals not necessarily equal in length. i 3 1 2 n Area of the 1 st rectangle: f ( 1 ) Δ 1 x 2 nd rectangle: f ( 2 ) Δ 2 x 3 rd rectangle: f ( 3 ) Δ 3 x f ( 1 ) = f ( 2 ) f ( 3 ) f ( i ) f ( n ) cdcjaurigue
R Δ 1 x Δ 2 x Δ 3 x Δ n x ε i 3 1 2 n c The area of R is approximated by the Reimann Sum ( ) ( ) ( ) ( ) ( ) 1 1 2 2 3 3 1 n n n i i i f x f x f x f x f x = ε Δ + ε Δ + ε Δ + ε Δ = ε Δ x i Δ ( ) i f ε The geometric interpretation of the Riemann sum is the sum of the areas of the rectangles with width and length Δ i x f ( 1 ) f ( 3 ) f ( i ) f ( n ) cdcjaurigue

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c The area of R is equivalent to ( ) 1 n i i i f x = ε Δ R Δ 1 x Δ 2 x Δ 3 x Δ n x ε i 3 1 2 n Riemann sum: ( ) 1 lim n i i n i f x →∞ = ε Δ cdcjaurigue
Definition . If no matter how we subdivide and no matter which numbers ’s we choose in each subinterval, exists and is equal to a real number L , we say that f is integrable on and the number L is called the definite integral of f on . We denote this by writing [ ] b , a i ε [ ] b , a [ ] b , a ( ) 1 lim n i i n i f x →∞ = ε Δ ( ) ( ) 1 lim n b i i a n i f x dx f x L →∞ = = ε Δ = cdcjaurigue

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Theorems on Definite Integrals
Theorem 1 The Fundamental Theorem of Calculus Let the function f be continuous on the closed interval and F be an antiderivative of f on the interval . Then [ ] b , a ( ) ( ) ( ) - = b a a F b F dx x f [ ] b , a cdcjaurigue

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c Theorem 2 ( ) ( ) - = a b b a dx x f dx x f ( ) a f ( ) 0 = a a dx x f ( ) ( ) = b a b a dx x f k dx x kf c Theorem 3 If is defined, c Theorem 4 for any real constant k . cdcjaurigue
c Theorem 5 ( ) ( ) ( ) ( ) b b b a a a f x g x dx f x dx g x dx ± = ± ( ) b , a c c Theorem 6 for each . ( ) ( ) ( ) b c b a a c f x dx f x dx f x dx = + cdcjaurigue

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Remark If a function F is defined at each number in , then [ ] , a b ( ) ( ) ( ) b a F x F b F a = - cdcjaurigue
Example 1. Evaluate ( ) ( ) ( ) b a F x F b F a = - ( ) 3 2 1 3 5 1 . x x dx - + - Solution: ( ) 3 2 1 3 5 1 x x dx - + -

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m23 definite integrals and area of plane region - The...

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