Use Eq .(4) to compute the arc length of y=ln((e^x+1)/(e^x-1)) over [1,3] .

Name: Problem 28, Chapter 8: Calculus for AP
Uploaded: 2020-09-27T20:05:34-08:00
Duration: 11 min 12 s
Channel: Melvin Adkins
Description: Use Eq .(4) to compute the arc length of y=ln((e^x+1)/(e^x-1)) over [1,3] .

Use Eq .(4) to compute the arc length of...

Key Concepts

Integration Techniques for Radical Functions

Integration techniques for dealing with radical functions become important when evaluating the arc length formula. The presence of the square root of an expression involving the derivative can complicate the integral, often necessitating substitutions or simplification methods to evaluate or approximate the integral. Mastery of these integration techniques is essential for successfully computing the exact or approximate length of a curve.

Differentiation and the Chain Rule

Differentiation, particularly using the chain rule, is essential when working with composite functions such as logarithmic expressions or ratios of exponential functions. This concept is crucial because finding the derivative of the function is the first step towards computing the integrand in the arc length formula. Understanding how to differentiate complex expressions ensures the correct formulation of the integrand in terms of the variable of integration.

Arc Length Formula

The arc length formula is a key concept in differential calculus and geometry, used to calculate the length of a curve over a specific interval. It is expressed as the integral of the square root of 1 plus the square of the derivative of the function with respect to x. This formula provides a bridge between differentiation and integration by requiring the computation of the derivative of the given curve followed by the evaluation of an integral.

Transcript

00:01 In this problem, we're going to compute the arc length of natural log of e to the x plus 1 over e to the x minus 1 from 1 to 3.

00:11 And recall that the arc length of natural log of f of x is equal to, in this case the integral from 1 to 3 of the square root of f of x squared plus f prime of x squared all over f of x of x.

00:45 D x where f of x is this inside function so let's compute f of x squared plus f prime of x squared well we know f of x which is just e to the x plus 1 over e to the x minus 1 so we can just square that that's e to the x again that's e to the x plus 1 over e to the x minus 1 but also let's multiply the top and bottom by e to the x minus 1 over e to the x minus 1.

01:55 And then square that and we'll see y in a moment.

01:59 So we have f of x squared, which is, this is essentially just f of x squared because this portion is just one.

02:09 But at any rate, we get f of x squared plus f prime of x squared.

02:14 So now we need to find the derivative of e to the x plus 1 over e to the x minus 1.

02:21 And we can find that using quotient rule.

02:26 So f prime of x is just the quantity e to the x minus 1 times the derivative of e to the x plus 1, which is e to the x minus the quantity e to the x plus 1 times the derivative of e to the x plus 1 times the derivative of e to the x plus 1, the x minus 1, which is e to the x, all over the quantity, e to the x minus 1 squared.

03:16 And then this simplifies to minus 2 times e to the x all over the quantity e to the x minus 1 squared.

03:34 So when we square f prime of x, which is, of course, this piece, when we square that, we get, we get that this plus 4 times e to the 2x all over the quantity e to the x minus 1 and then this is to the 4th power.

04:06 All right, so now that we have that, what we want to do is simplify f squared plus f prime squared.

04:19 So we get that this is equal to, we can write this portion, e to the x plus 1, times e to the x minus 1.

04:32 That's just e to the 2x minus 1 and that's still squared.

04:47 And this is plus 4 e to the 2x all over e to the x minus 1 to the 4th power.

05:04 And then we can write this product out so we get that this is equal to that would be e to the 4x minus 2 times times e to the 2x plus 1 plus 4 times e to the 2x.

05:35 Again, this is all over.

05:39 E to the x minus 1 to the 4 power.

05:45 And then this is equal to e to the 4x plus 2 times e to the 2x plus 1, which we can write as a quantity e to the 2x plus 1...

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