13. Briefly explain the difference between taking a left sum and the limit of a left sum as the number of rectangles goes to infinity in regard to finding the area under a curve (as discussed in class).

Answered: 13. Briefly explain the difference…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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13. Briefly explain the difference between taking a left sum and the limit of a left sum as the number of rectangles goes to infinity in regard to finding the area under a curve (as discussed in class).

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Step 1

$R i e m a n n s u m The given domain, x ∈[a,b] is sub divided into 'n' small intervals of width∆x. ∆ x = \frac{b - a}{n} {Starting from x}_{i} w h e r e i = 0, 1, 2, . . ., n - 1, n . S = \sum_{i = 1}^{n} f (x_{i}^{*}) ∆ x_{i} F o r L e f t s u m : x_{i}^{*} = x_{i - 1} F o r R i g h t s u m : x_{i}^{*} = x_{i}$

Step 2

$F o r i n c r e a s i n g f u n c t i o n F o r l e f t s u m : T h e h e i g h t o f r e c \tan g l e, f (x_{i - 1}) i s e q u a l t o f (x) a t l e f t e n d o f t h e r e c \tan g l e . B u t f o r x_{i - 1} t o (x_{i - 1} + ∆ x), t h e h e i g h t o f r e c \tan g l e i s a l w a y s l o w e r t h a n f (x) . H e n c e t h e a r e a c o v e r e d o r c a l c u l a t e d u n d e r t h i s s u m i s l o w e r t h a n t h e a c t u a l a r e a u n d e r f (x) c u r v e . F o r R i g h t s u m : T h e h e i g h t o f r e c \tan g l e, f (x_{i}) i s e q u a l t o f (x) a t R i g h t e n d o f t h e r e c \tan g l e . B u t f o r x_{i} t o (x_{i} - ∆ x), t h e h e i g h t o f r e c \tan g l e i s a l w a y s h i g h e r t h a n f (x) . H e n c e t h e a r e a c o v e r e d o r c a l c u l a t e d u n d e r t h i s s u m i s h i g h e r t h a n t h e a c t u a l a r e a u n d e r f (x) c u r v e .$

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