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Use the properties of integrals to verify the ine…

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Problem 56 Easy Difficulty

Use the properties of integrals to verify the inequality without evaluating the integrals.

$ \displaystyle \int^1_0 \sqrt{1 + x^2} \,dx \le \int^1_0 \sqrt{1 + x} \,dx $


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Frank Lin

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Amrita Bhasin

Related Courses

Calculus 1 / AB

Calculus: Early Transcendentals

Chapter 5

Integrals

Section 2

The Definite Integral

Related Topics

Integrals

Integration

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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.

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40:35

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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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Video Transcript

all right, we're going to try to calculate or prove this inequality that you see right here. What's important is that the book specifically asks you to do it without evaluating the integral. So how do we approach this? We use one of the properties of Integral Is that says if you have the same limit of integration and if you have an expression that's less then the other function than the integration is also going to be less than it. So long story short, we're going to prove that this portion is larger than that portion and that doesn't involve the evaluation of thes integral. So we're going to be able to do that. Okay, so let's compare the square root off one plus X squared versus squared of one plus X. We know that square root functions are 1 to 1. So we know that the input is what really matters here to decide which ones larger. So I'm gonna look at X squared versus X. I am going to ignore the one because one is just added to both of them. So if I subtract it from one, you can subtract it for the other, and then the relationship between which ones larger and which one smaller doesn't change. Okay, now X squared versus X. Just way looking at it. You might think that X squared could be larger than X. But that's not necessarily true, because in this case, we're looking at the region 0 to 1. Okay, graphically, what's happening is this X squared looks like this. Well, X looks like that between the region 0 to 1, so eventually X squared is going to be larger than X. But when X is a number between 0 to 1, we know that X is actually larger than X square. So what's the relationship that we see right here? X squared is less than X and going going back, we can show that the square root off one plus x squared iss less than the square root off one plus X and according to the properties of Integral, this is also going to be too. Now, what's important is the fact that it's from 0 to 1. Okay, Without this, we're not able to prove this portion, so it wouldn't work. So, keeping that in mind, we were able to prove this situation

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Top Calculus 1 / AB Educators
Grace He

Numerade Educator

Catherine Ross

Missouri State University

Kayleah Tsai

Harvey Mudd College

Joseph Lentino

Boston College

Calculus 1 / AB Courses

Lectures

Video Thumbnail

05:53

Integrals - Intro

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.

Video Thumbnail

40:35

Area Under Curves - Overview

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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