Arc length Use the result of Exercise 108 to find the arc leng...

Question

Arc length Use the result of Exercise 108 to find the arc length of the curve: f(x) = ln |tanh(x / 2)| on [ln 2, ln 8].

Accepted Answer

Hello there. Today we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Find the arc length of the curve G and Z is equal to the natural log of the absolute value of hyperbolic tangent of. Z divided by 2. On the interval, Z is an element of the closed interval of the natural log of 3, the natural log of 27. Awesome. So it appears for this particular problem we're asked to find the arc length of this curve G of Z on this specific interval given the information that is provided to us by the prom itself. So with that in mind, now that we're all know that we're ultimately trying to solve for this arc length of this curve on this specific interval. Let us first devise a strategy to help us solve for this problem. So as we should recall a note, the problem statement asks us to identify the arc length of a particular curve. For that purpose, we will utilize the arc length formula which we should recall to be. L is equal to the integral from Z1 to Z2 of the square root of drum roll, 1 + parenthesis. Of Z to the power of 2 D Z. Fantastic. The strategy to solve this particular problem relies on identifying the limits of integration from the interval, and we need to then determine what the derivative is, and then we also need to define the integrand and then we will need to integrate. Fantastic. So with that in mind, now that we have a strategy in place and that we've recalled and used the arc length formula, our first step that we need to take to solve this problem is we need to focus our attention on the limits of integration. So according to the prompt statement, as we should recall, the lower limit of integration, which I'll call LL for short, the lower limit of integration is Z1 is equal to the natural log of 3, while the upper limit, which I'll call UL upper limit. So, the upper limit of integration corresponds to Z2 is equal to the natural log of 27. And once again, since we can see from the problem itself that our interval is given to us in closed brackets, so it's going to be a closed interval because it's given to us in square brackets. And because it's in square brackets, once again these are end points of a closed interval. Fantastic. So with that in mind, our second step is we need to perform differentiation. So in order to find the derivative of G of Z, we are going to have to recall and use the chain rule since we have 1 outer and two inner functions. For this particular part. There is no necessity to explain everything in detail as it is part of concepts of derivatives, so we need to primarily focus on techniques of integration for this particular problem. So that will mean that one of the prerequisites required in order to perform this derivative is we need to recall that, and let's write this in blue, let us recall that. The natural log prime of the absolute value of X is equal to 1 divided by X. So with that in mind, let us perform the differentiation to find the derivative of G of Z. So G Z, so G Z. is equal to D by Dz of the natural log of the absolute value of hyperbolic tangent of Z divided by 2. is equal to D divided by D multiplied by parenthesis of hyperbolic tangent of Z divided by 2. Multiplied by the natural log of The absolute value of hyperbolic tangent of Z divided by 2. Multiplied by drumroll, D divided by D multiplied by parenthesis, Z divided by 2, so Z divided by 2. Fantastic. Multiplied by hyperbolic tangent of Z divided by 2 multiplied by D by Dz of. So it's gonna be D by DZ or D divided by DZ multiplied by Z divided by 2. Fantastic. Moving right along here. Then we can go ahead and write that G of Z, so G Z is equal to 1 divided by the hyperbolic tangent of Z divided by 2 multiplied by the hyperboliccant, so hyperbolic. C can't squared of Z divided by 2 multiplied by 1/2. And continuing on with our solving, we can say that G of Z is equal to the square root of hyperbolic cosine of Z divided by 2, so Z divided by 2. Divided by hyperbolic sign of Z divided by 2, multiplied by 1 divided by hyperbolic cosine. To the power of The square root of 2. Of Z divided by 2 multiplied by 12, which will mean that G of X will be equal to 1 divided by 2 multiplied by hyperbolic sign of Z divided by 2 multiplied by hyperbolic cosine. Of Z divided by 2. And from this point, we need to recall and use the double angle identity, which we should recall as, and I'll write this in blue as hyperbolic sign, so we need to take hyperbolic sign of 2 Z is equal to 2, it's gonna be equal to 2. So Gonna be equal to 2 multiplied by the hyperbolic sign of Z of Z multiplied by the hyperbolic cosine of Z. Fantastic. So that will mean, as we continue on with our solving, that G of X will be equal to 1 divided by hyperbolic sine of Z will be equal to. Cosicant, hyperbolic cosicant to be precise, hyperbolic cos he can't. Of Z. Fantastic. So, with that in mind, so now we can move on to our 3rd step. So moving right along here to our 3rd step. So step number 3, we now need to define the integrand. So defining our integrand. So in order to do that, we need to use the derivative found in step 2, and we need to show that the square root of 1 + parenthesis. Of Z So the power of 2. will be equal to the square root of one plus. So, once again, we're gonna be taking the hyperbolic cosicant, so hyperbolic cosicant square of Z, which will be equal to the square root of the hyperbolic cotangent squared of Z, which will give us an answer of plus or minus. Hyperbolic cotangent of Z. Fantastic. So, with that in mind, since Z is an element of the close interval of the natural log of 3, the natural log of 27, we also need to note that Z is greater than 0, and we can take the positive solution as Hyperbolic cotangent of Z is greater than 0. And that Z is greater than 0. So that will mean that the integrand, therefore, without a doubt our integrand will be, and let's write this in black, our integrand will be hyperbolic cotangent of Z. Fantastic, and that is our integrant. So with that in mind, our fourth step now is we need to perform integration. So we wish to integrate L is equal to the integral from the natural log of 3 to the natural log of 27 of hyperbolic cotangent of ZD Z. Sense and we also need to note that sense, and let's write this in blue, since hyperbolic cotangent of Z is equal to Hyperbolic cosine of Z divided by hyperbolic sine. So hyperbolic sign of Z. So let us state that you will be equal to hyperbolic sign of Z. Fantastic. And then the derivative, so let's make a note that the derivative DU divided by DX will be equal to hyperbolic cosine of Z. And the numerator of the Integrand. So let us know that the numerator of the integrand will be the interval of Hyperbolic cotangent of Z D Z will be equal to the interval of hyperbolic cosine of Z divided by hyperbolic sign of Z. D Z. Fantastic. So that will mean that DU, once again, let's make a note in blue, that DU will become equal to hyperbolic cosine of ZD Z. Fantastic. So now we need to find the new limits of integration. And with that in mind, let's start with U1 and U1 is equal to hyperbolic sign of Z1 is equal to hyperbolic sign of the natural log of 3, which will be equal to E to the power of the natural log of 3 minus E to the power of negative natural log of 3. Divided by 2. And when we simplify this expression down to its most simple form, we will find that it's equal to 4 divided by 3. Moving on to U2, U2 will be equal to hyperbolic sign of Z2, which will be equal to hyperbolic sign of the natural log of 27. Which will be equal to E to the power of the natural log of 27 minus E to the power of the natural log of 1 divided by 27. Fantastic, divided by 2. And when we simplify this expression down to its most simple form, we will find that it's simply equal to 364 divided by 27. Fantastic. So with that in mind, the integral will become L is equal to the integral of 4 divided by 3 to 364 divided by 27 of. DU divided by U, which is equal to the natural log of you evaluated from 4 divided by 3 to 364 divided by 27. And now our next step is we need to recall and apply the evaluation theorem, and when we apply the evaluation theorem, we can obtain the final answer, so we will find that L is equal to the natural log of 364 divided by 27 minus the natural log of 4 divided by 3. Which will be equal to the natural log of 364 divided by 27 multiplied by 3 divided by 4, which will give us an answer of the natural log of 91 divided by 9, and that's it. That is our final answer. Hooray, we did it. So with that in mind, let's recap what we've learned and look at the problem itself. So that will mean that based on everything that we've learned together, that our final answer without a doubt, will have to be the natural log of 91 divided by 9. And this is our final answer. Hooray, we did it. So therefore, without a doubt, in conclusion, the arc length of the curve over the provided interval will have to be the natural log of 91 divided by 9. And that's it. Thank you so much for watching. Hopefully that helped, and I can't wait to see you in the next video. Bye.

Arc length Use the result of Exercise 108 to find the arc length of the curve: f(x) = ln |tanh(x / 2)| on [ln 2, ln 8].

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