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Problem 22 Easy Difficulty

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 $

Answer

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

Discussion

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CA

Catherine A.

October 27, 2020

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

SA

Sharieleen A.

October 27, 2020

This will help a lot with my midterm

ON

Olivera N.

November 3, 2020

why is it 9??

ak

Abdulaziz K.

September 13, 2021

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

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