Find-the-Riemann-sum-for-the-given-function-with-the-specified-number-of-intervals-using-left-endpoints-f-x-4ln-x-2x-1-x-4-n-7-Round-your-answer-to-two-decimal-places-

Question Number 125997 by bramlexs22 last updated on 16/Dec/20

F i n d t h e R i e m a n n s u m f o r t h e

g i v e n f u n c t i o n w i t h t h e s p e c i f i e d

n u m b e r o f i n t e r v a l s u s i n g l e f t

e n d p o i n t s f (x) = 4 \ln x + 2 x; 1 ⩽ x ⩽ 4

n = 7 . R o u n d y o u r a n s w e r t o t w o

d e c i m a l p l a c e s ?

Answered by liberty last updated on 16/Dec/20

W e w a n t t o u s e s e v e n r e c t a n g l e s t o e q u a l s

w i d t h t o e s t i m a t e t h e a r e a u n d e r f (x) = 4 \ln x + 2 x

o v e r t h e i n t e r v a l 1 ⩽ x ⩽ 4 .

Δ x = \frac{4 - 1}{7} = \frac{3}{7}

t h e n d e v i d e t h e i n t e r v a l [1, 4] i n t o 7 e q u a l s

p i e c e s . G i v e s x = 1, x = \frac{10}{7}, x = \frac{13}{7}, \dots, x = \frac{25}{7}

t h e a r e a s : \frac{3}{7} f (1) + \frac{3}{7} f (\frac{10}{7}) + \frac{3}{7} f (\frac{13}{7}) + \dots + \frac{3}{7} f (\frac{25}{7})

= \frac{3}{7} {4 \ln 1 + 2 + 4 \ln \frac{10}{7} + \frac{20}{7} + 4 \ln \frac{13}{7} + \frac{26}{7} + \dots + 4 \ln \frac{25}{7} + \frac{50}{7}}

\approx 22.66

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