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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 $

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01:36

Frank Lin

00:47

Amrita Bhasin

05:45

Linda Hand

01:31

Carson Merrill

Calculus 1 / AB

Chapter 5

Integrals

Section 1

Areas and Distances

Integration

Abdulaziz K.

September 13, 2021

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

Olivera N.

November 3, 2020

why is it 9??

Sharieleen A.

October 27, 2020

This will help a lot with my midterm

Catherine A.

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

Missouri State University

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

Lectures

05:53

In mathematics, an indefinite integral is an integral whose integrand is not known in terms of elementary functions. An indefinite integral is usually encountered when integrating functions that are not elementary functions themselves.

40:35

In mathematics, integration is one of the two main operations of calculus, with its inverse operation, differentiation, being the other. Given a function of a real variable (often called "the integrand"), an antiderivative is a function whose derivative is the given function. The area under a real-valued function of a real variable is the integral of the function, provided it is defined on a closed interval around a given point. It is a basic result of calculus that an antiderivative always exists, and is equal to the original function evaluated at the upper limit of integration.

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Use Definition 2 to find a…

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$15-17$ Use Definition 2 t…

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$17-19$ Use Definition 2 t…

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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