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 ?


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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

	$W e w a n t t o u s e s e v e n r e c t a n g l e s t o e q u a l s$ $w i d t h t o e s t i m a t e t h e a r e a u n d e r f (x) = 4 \ln x + 2 x$ $o v e r t h e i n t e r v a l 1 ⩽ x ⩽ 4 .$ $Δ x = \frac{4 - 1}{7} = \frac{3}{7}$ $t h e n d e v i d e t h e i n t e r v a l [1, 4] i n t o 7 e q u a l s$ $p i e c e s . G i v e s x = 1, x = \frac{10}{7}, x = \frac{13}{7}, . . ., x = \frac{25}{7}$ $t h e a r e a s : \frac{3}{7} f (1) + \frac{3}{7} f (\frac{10}{7}) + \frac{3}{7} f (\frac{13}{7}) + . . . + \frac{3}{7} f (\frac{25}{7})$ $= \frac{3}{7} {4 \ln 1 + 2 + 4 \ln \frac{10}{7} + \frac{20}{7} + 4 \ln \frac{13}{7} + \frac{26}{7} + . . . + 4 \ln \frac{25}{7} + \frac{50}{7}}$ $\approx 22.66$
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