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Lecture

Notes Integrals

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Department
Mathematics
Course
MATH 140
Professor
All Professors
Semester
Fall

Description
Notes – Integrals Aleah Pisarz 10/29 – 11/7  1 Riemann Sums: Riemann Sums are used to take the area under the curve without using integrals. When  P=x ,0 ,1 ,2 ,…3 The Lower Riemann Sum can be shown by L(P,f )=f (L) x −x +f (L) x −x +…+f (L)(x −x ) ( 1 0) ( 2 1) n n−1 where  L  is the minimum of the area defined by  xn−x n−1 The Upper Riemann Sum can be shown by U (P,f )=f (U )(x1−x 0) (U ) ( 2x +…1f (U)(x −x n n−1) where ? is the maximum of the area defined by  xn−x n−1 Ex. 1+x,0≤x≤2 f x = { 5−x,2≤ x≤4 P= {0,1,3,4 } L(P,f )=¿ 2 Notes – Integrals Aleah Pisarz 10/29 – 11/7  U (P,f )=¿ Fundamental Theorem of Calculus: If  f (x)  is continuous on  [a,b] , differentiable (a,b) ,  and f x −g(x) , then b ∫ g(x)dx=f (b)−f (a) a U­substitution: 1. Let  u  = what’s in parenthesis or under root symbols 2. Take deriv
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