Find the length of the following polar curves. The spiral r=2 e^2 θ, for 0 ≤θ≤ln8

Name: Problem 69, Chapter 12: Calculus: Early Transcendentals
Uploaded: 2020-08-01T01:10:34-08:00
Duration: 5 min 15 s
Channel: Victoria Dollar
Description: Find the length of the following polar curves. The spiral r=2 e^2 θ, for 0 ≤θ≤ln8

Find the length of the following polar curves. The spiral...

Key Concepts

Integration Techniques

Evaluating the arc length integral often involves techniques for handling exponential functions and other complex integrals. Knowledge of substitution and simplification strategies is critical to successfully compute the integral, which ultimately yields the length of the curve.

Polar Coordinates

Polar coordinates represent points in the plane with a radius and an angle, which can simplify the description and analysis of curves that exhibit circular or spiral symmetry. This system is especially useful when the curve is naturally defined in terms of angles and distances from the origin.

Arc Length Formula for Polar Curves

The arc length of a curve in polar coordinates, where the curve is given by r = f(?), is determined by the integral L = ?[a, b] ?((dr/d?)² + r²) d?. This formula accounts for the change in both the radius and the angle, and it is derived by translating the standard arc length concept from Cartesian to polar coordinates.

Differentiation and the Chain Rule

In the context of functions involving exponentials or other composite functions, differentiation using the chain rule is essential. It allows for the correct computation of dr/d?, which is required for evaluating the arc length integral in polar coordinates.

Transcript

00:01 Okay, so for this problem, we're asked to find the arc length of r equals 2e to the 2 pi.

00:06 So for the length formula, we're going to use the integral of the interval a to b, and then the square root of the function of theta squared, plus the derivative of the function squared, and all of that integrated in terms to theta.

00:31 So for this one we have zero and natural log of a of eight being our ab.

00:42 And then r is our function of theta.

00:45 So the only thing we're needing at this point is the derivative.

00:49 So r prime is going to equal two.

00:53 And then the derivative of e to the two theta power is e to the two theta times two, which is four e to the two theta.

01:06 So i'm going to go ahead and set up my integral.

01:12 So from 0 to natural log of 8, and then it's going to be 2e to the 2 theta squared, plus 4e to the 2 theta squared, and then d theta.

01:37 So the first thing i'm going to do is i am going to simplify these down.

01:45 I'm going to go ahead and square everything.

01:46 So 2 squared is 4, and then e to the 4 theta, because we're going to do 2 theta times 2, which is 4.

01:57 And then 4 squared is 16 and then again e to the 4 theta.

02:08 These are going to add together to be 20, e to the 4 theta.

02:23 And then what i'm going to do is i'm going to go ahead and separate out my 20 and i'm going to go ahead and rewrite my e to the 4 theta.

02:46 So i can rewrite this as e to the 2 theta to the 2 power.

02:54 And what this is going to allow me to do is it's going to allow me to go ahead and take it out of the radical.

03:04 So now it's just going to be e to the 2 theta because i have two of them multiply together...

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