Refer a friend and earn $50 when they subscribe to an annual planRefer Now

Get the answer to your homework problem.

Try Numerade Free for 30 Days

Like

Report

Find the length of the curve $$y=\ln \left(1-x^{2}\right), 0 \leq x \leq \frac{1}{2}$$

$\frac{-1}{2}+\ln 3$

Calculus 1 / AB

Calculus 2 / BC

Chapter 8

Techniques of Integration

Section 5

Integration of Rational Functions by Partial Fractions

Integration

Integration Techniques

Campbell University

University of Michigan - Ann Arbor

Boston College

Lectures

01:11

In mathematics, integratio…

06:55

In grammar, determiners ar…

06:02

Find the exact length of t…

08:12

03:13

17:38

Find the length of the cur…

22:04

04:37

04:53

03:36

07:02

02:34

problem number 73. So we're dealing with problems that requires to be able to do partial fraction decomposition. Here they ask us to find the length of a curve along the length along the curve, like with the natural log of one money's X squared for X values between zero and 1/2. So we're guided by our formula for art went over An interval is going to be the square root of one plus the derivative of that function squared. So what we need to do in this case, we're looking at a function f of X is equal to the natural log of one minus x square. So what is the derivative of that function? The derivative is going to be 1/1 minus X squared and then using the chain roll, the derivative of one minus X squared is what, minus two X And so this case you've got minus two X over one minus X squared. And if you just sort of like to not deal with that minus sign, I could just write. This is two X over X squared minus one. So that is by derivative. Now I just need to plug that into the formula that we have. So I'm gonna need to figure out then what is the square root of one plus f prime of X square? So I need to work on one plus what we say. The derivative, the derivative was two x over X squared, minus one squared. So now let's just get a common denominator here. The common denominator is X squared, minus one squared. So you're going tohave X squared, minus one squared plus two x squared. So this is the square root of X to the fourth minus two X squared plus one plus four X squared over X squared minus one squared, which is the square root of that's X to the fourth plus two X squared plus one over X squared minus one squared, which is X square plus one squared over the square root of X squared minus one squared. So what I've got at this point, I've got that the art length the art link that I'm looking for is going to be the integral from zero toe 1/2 of the square root of X square plus one squared over X squared minus one squared DX. So this is the integral from 0 to 1/2 absolute value X squared plus one squared minus one DX. Now, on the interval from 0 to 1/2 you know that the numerator is going to be a positive numbers. You got a positive number plus a positive number. So this turns into the integral from zero to 1/2. I could just get rid of the absolute values there. Now, if you look at the denominator on the integral from 0 to 1/2 that number is going to start it. When I plug into zero, it's a negative one. When I plug in a 1/2 is a 1/4 minus one. That is always a negative number to take the absolute value of a negative number. You just take the opposite of that number. So X squared minus one DX. So this is, um, minus the integral from 0 to 1/2 of x square, plus one of rest squared minus Wendy X. So that is what we're tasked with. Now it's finding the interval from zero minus from 0 to 1/2 of X squared, plus one of Rex, where mine is one DX. Now I'm gonna leave you to do this. If you've gotten this far in these exercises, you should be quite comfortable with doing partial fraction decomposition. So X squared plus one over X squared minus one. It's simply going to be one minus one over X plus one plus one over X minus point. That is the partial fraction decomposition, and you can go through the process to end up with that. So what that tells us is I need the integral from 0 to 1/2 off, one minus one over X plus one was one of Rex minus one DX, and now all of this easily integrates. So this is going to be minus when you innovate one who just going to get what X and then minus natural log X plus one plus natural log X minus one and then all this. It gets evaluated from zero to 1/2. So let's go do that on the next page. So I've got minus X minus natural log of X plus one plus natural logger X minus one and this evaluated from 0 to 1/2 so minus X, Um, and that you could just write this plus, um, natural log X minus one over X plus one from 0 to 1/2 and that this is now. We just turn this into arithmetic problem to get to our final answer for the argument. So let's do the substitution. Said this is going to be minus some quantity minus minus some quantity. So let's substitute a 1/2 when you substitute a 1/2 and x, you get 1/2 and this is plus the natural log of so 1/2 and over three halves. Typically, the minus sign is going to get caught in the absolute value. When you substitute a zero in, you're going to get a zero and then you're going to get plus natural log at survey of minus one over one. So you know that this term is zero. This term is zero. And so what you end up with is minus 1/2 and what you end up with in this case right here, you've got 1/2 over three halves, so this is just 1/3. So this is plus, So is the natural log of 1/3 which is minus. You got 1/2 you got plus natural law go one minus natural log of three natural Aga one is simply zero said This final answer is minus 1/2 plus the natural log of three. So it's quite a bit of ways, but that is the art lent. So for this curve from 0 to 1/2 the art length is natural. Log of three minus one halfs again. What we did was usar art length formula. The art lent formula lead us along a path that where we needed to know what a partial fraction decomposition is. Once we get the partial fraction decomposition, the integration becomes straightforward, and they just substitute the value 0 1/2 to get art length of that curve.

View More Answers From This Book

Find Another Textbook

Numerade Educator

In mathematics, integration is one of the two main operations in calculus, w…

In grammar, determiners are a class of words that are used in front of nouns…

Find the exact length of the curve.$$y=\ln \left(1-x^{2}\right), \quad 0…

Find the exact length of the curve.$$y=1-e^{-x}, \quad 0 \leqslant x \le…

Find the exact length of the curve.

$ y = \ln (1 - x^2) $ , $ 0 \le …

Find the length of the curve $y=1-e^{-x}, 0 \leq x \leq 1$

Find the length of the curve $y=1-e^{-x}, 0 \leq x \leq 1$.

Find the exact length of the curve.$$y=\frac{1}{4} x^{2}-\frac{1}{2} \ln…

Find the length of the curve $y=e^{x}$ over $[0, \ln (2)]$

$ y = 1 - e^{-x} $ , $ 0 \le x \…

Find the exact length of the curve.$$y=\ln (\sec x), \quad 0 \leqslant x…

08:08

The instructions for the integrals in Exercises $1-10$ have two parts, one f…

03:23

Use the Euler method with $d x=0.5$ to estimate $y(5)$ if $y^{\prime}=$ $y^{…

00:53

In Exercises $21-42,$ find the derivative of $y$ with respect to the appropr…

07:10

In Exercises $27-40$ , use a substitution to change the integral into one

02:00

Evaluate the integrals in Exercises $39-54$$$\int \frac{(x+1)^{2} \t…

01:22

In Exercises $35-68$ , use integration, the Direct Comparison Test, or the L…

01:38

Find the values in Exercises $9-12$$$\cot \left(\sin ^{-1}\left(-\fr…

Evaluate the integrals in Exercises $39-54$$$\begin{array}{l}{\int \…

01:34

01:45

Use reference triangles like those in Examples 1 and 3 to find theangles…

92% of Numerade students report better grades.

Try Numerade Free for 30 Days. You can cancel at any time.

Annual

0.00/mo 0.00/mo

Billed annually at 0.00/yr after free trial

Monthly

0.00/mo

Billed monthly at 0.00/mo after free trial

Earn better grades with our study tools:

Textbooks

Video lessons matched directly to the problems in your textbooks.

Ask a Question

Can't find a question? Ask our 30,000+ educators for help.

Courses

Watch full-length courses, covering key principles and concepts.

AI Tutor

Receive weekly guidance from the world’s first A.I. Tutor, Ace.

30 day free trial, then pay 0.00/month

30 day free trial, then pay 0.00/year

You can cancel anytime

OR PAY WITH

Your subscription has started!

The number 2 is also the smallest & first prime number (since every other even number is divisible by two).

If you write pi (to the first two decimal places of 3.14) backwards, in big, block letters it actually reads "PIE".

Receive weekly guidance from the world's first A.I. Tutor, Ace.

Mount Everest weighs an estimated 357 trillion pounds

Snapshot a problem with the Numerade app, and we'll give you the video solution.

A cheetah can run up to 76 miles per hour, and can go from 0 to 60 miles per hour in less than three seconds.

Back in a jiffy? You'd better be fast! A "jiffy" is an actual length of time, equal to about 1/100th of a second.