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# Use Definition 2 to find an expression for the area under the graph of $f$ as a limit. Do not evaluate the limit.$f(x) = x^2 + \sqrt{1 + 2x}, \hspace{5mm} 4 \le x \le 7$

## Area $=\lim _{n \rightarrow \infty} \frac{3}{n} \sum_{i=1}^{n}\left(\left(4+\frac{3 i}{n}\right)^{2}+\sqrt{9+\frac{6 i}{n}}\right)$

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Catherine A.

October 27, 2020

I've been struggling with Calculus: Early Transcendentals, this is so helpful

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Sharieleen A.

October 27, 2020

This will help a lot with my midterm

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Olivera N.

November 3, 2020

why is it 9??

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Abdulaziz K.

September 13, 2021

Determine a region whose area is equal to the given limit. Do not evaluate the limit.

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### Video Transcript

for this problem, we need to find an expression for the area under the graph of F of X, which is equal to X squared plus the square root of one plus two X over the interval 47 as a limit. And using only definition to a definition to states that the area this is equal to the limit is and approaches infinity of delta X times is a mission I from one to end of F of X sub I where delta X is just the difference between the end points A and B. So that's B minus A over the number of sub intervals. In now we let except by be right endpoints of these sub intervals. Then if A is the left most endpoint we have except by this is equal to A plus I times delta X. And so we write area equal to limit as N approaches infinity of delta X times information I from one to end of F of a plus delta X times I. Now, if f of X is X squared plus the square root of one plus two X over the interval 47 then we have, you say delta X. This is equal to b minus a over end, which is just seven minus four over and or three over in. And we get area, this is equal to the limit as N approaches infinity of delta X, which is three over in times the summation I from one to end F of a S four plus I times three over in. And so replacing X by four plus I times three over in. We have area which is equal to the limit as an approaches infinity of three over in times it's a mission I from one to end of we have four plus three I over end squared plus the square it of one plus two times four plus three I over. In. And so this is the area in terms of definition to

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