Download the App!

Get 24/7 study help with the Numerade app for iOS and Android! Enter your email for an invite.

Get the answer to your homework problem.

Try Numerade free for 7 days

Like

Report

Determine the area of the indicated region.Region bounded by $f(x)=2 / x$ and the $x$ -axis, between $x=-3$ and $x=-1$.

$$2 \ln 3$$

Calculus 1 / AB

Chapter 5

Integration and its Applications

Section 6

The Definite Integral

Integrals

Campbell University

Oregon State University

Harvey Mudd College

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.

01:52

Determine the area of the …

01:32

03:06

01:44

01:29

02:25

14:06

Find the area of the regio…

04:20

right. So this is a interesting problem with area. A lot of people associate area as always, being positive. Um, so if you're in that category of saying the area is always positive that you want to make sure that you do the absolute value of the problem. Um And when you see the X axis is telling you to do the integral of the function that they give you two over x dx and they give you the bounds from X equals negative 32 X equals negative one. Uh, now I like to go in order from least to greatest. But again, if you're considering area to be always positive and you can absolute value that answer, So if you get a negative, which I believe you will, um you just have to think Oh, okay. Well, area needs to be a positive value. So the derivative of natural log of X is equal to one over X. So for working backwards here, it makes sense that the anti derivative or the integral of this problem is to natural log of X. Now, notice that I'm putting an absolute value here because we can only log positive numbers If you're confused, why, it's two times we'll just think of it this way is you can move a constant in front and becomes two times one over X. And then you can apply this This rule, I guess, from negative 32 negative one. So, as we're looking at that problem, um, we're gonna plug in to natural your upper bound and being one minus two natural log. And again, I'm plugging in negative one. But the absolute value of next one is possible. And so I guess I did that already. Natural log of three. Now, this is where we're going to get into that debate on our answer because and I expect my students, you know, the natural log of one is zero and zero times. Anything is zero minus. That's going to give you a negative answer, and that's where we go back to. An area needs to be positive. So it's an absolute value. All of this, and you're left with just the positive answer, which would be to natural log of three, because two times 00 minus this is that answer. Um, but it has to be positive, so we're left with that

View More Answers From This Book

Find Another Textbook

Numerade Educator

02:05

Evaluate the given integral and check your answer.$$\int\left(4 x^{3}-9 …

01:43

Evaluate the given definite integral.$$\int_{-2}^{3}|t| d t$$

(a) Compute $\frac{d}{d x}(x \ln x-x)$. (b) What function can you now antidi…

02:26

(a) determine a definite integral that will determine the area of the region…

01:17

Determine the region $R$ determined by the given double integral.$$\int_…

01:18

04:12

Find (a) $f_{x x}(x, y),$ (b) $f_{y y}(x, y),$ (c) $f_{x y}(x, y),$ and $f_{…

Determine the area of the indicated region.Region bounded by $f(x)=\frac…

01:47

03:18

(a) determine the average value of the given function over the given interva…