How do you find the arc length of the polar curve r=f(θ),...

Key Concepts

Polar Coordinates

Polar coordinates use a radial distance and an angle to represent points in the plane. This framework is particularly advantageous for curves that naturally relate to rotations or symmetry about a point, allowing functions to be expressed in terms of the angle ? and a radius r that varies with ?.

Arc Length of a Curve

The arc length is the distance measured along the curve between two points. Determining the arc length involves summing up infinitesimal distances along the curve, which requires an expression for the differential element of length (ds) derived from the geometry of the curve.

Differential Element in Polar Coordinates

In the context of polar coordinates, the differential element of the curve's length combines changes in both the radius r and the angle ?. The formula for the differential arc length, ds = ?(r² + (dr/d?)²) d?, arises from applying the Pythagorean theorem to the infinitesimal displacement in the radial and angular directions.

Integration

Integration is the mathematical process used to accumulate the total arc length over a given interval. By integrating the expression for ds from the starting angle to the ending angle, one computes the complete length of the curve in polar coordinates.

Transcript

00:01 All right, hello again.

00:02 So we need to find out the arc length if we're given some polar curve, some function of f theta that represents rr, okay? and we're given this kind of unfun looking equation.

00:24 And essentially what's going on is that when we're polarizing things, x then becomes, if we want a nice picture.

00:35 Of it, we have some r, right? and when we have that r, let me try that draw that a little bit better, r, right? if we, from some angle theta, right, x is equal to r cost theta, and then the part up here, right, that's going to be r -y, and that's equal to r -syn of theta.

01:10 So this is definitely like i'm trying to make this visual for you guys so you can see it.

01:14 Now what's going on here is this is the change in x, right? so we go from some point x not.

01:23 Let me write like a little blue thing, x not to x one.

01:30 And at the same time we go from y1 or sorry, that should be y not or y zero to y1.

01:38 And these changes are expressed with these derivatives right here, which is why we have that.

01:46 So anyways, our task, i believe, is to rewrite this equation right there in a nicer form that involves this, the f theta.

02:02 So how do we get f theta in there? so there's a few things our book tells us, and it tells us that r is equal to some function, right? like we have that as is declared.

02:17 It's even declared right here, which is completely fine.

02:21 So let's try to rewrite our functions of x and y, these guys, with our f theta.

02:32 So if r is equal to f theta, then that tells us.

02:38 Us that x is then equal to f theta cosine of theta and then if we have y that's equal to f theta sine theta now if we want to take the derivative of that because we need to from our equation here let's go figure out what d x the change in x is with respect to theta so all we're going to do now is just do this.

03:22 And yes, we don't know what f of theta is, but that doesn't matter.

03:26 It's not going to stop us from being able to take a simple derivative.

03:30 We can take it in a generalized form that will be applicable to any function we would put in there for r.

03:37 So all this is going to be as a product rule.

03:40 So we're going to just going to get f prime of theta, post theta, plus f theta, plus f theta, cosine, actually this becomes minus, and that's sine of theta.

03:59 All right.

04:00 So i took the derivative of the first thing here and multiplied it by the second thing, right here.

04:07 So it gives us a derivative of f of theta, f prime, times cosine of theta.

04:13 And then what happened here was i left f of theta alone.

04:16 Then i took a derivative with respect to cosine, which the derivative of cosine is negative sine.

04:24 Of theta, right? now we have to do the same thing for d .y.

04:31 Sorry, the same thing for y.

04:33 So we take the derivative of respect to theta.

04:37 Let me rewrite that real quick.

04:40 There we go.

04:41 So it's going to be the same exact idea.

04:43 So to make a long story short, this is then going to be f of theta, sine of theta, plus when we take the derivative of sine of theta, that's just cosine of theta.

04:55 So then that's just going to be our regular prime notation, cosine of theta.

05:03 All right.

05:04 Now what we're going to do is we're going to take this expression.

05:06 We're going to go plug it back in to our expression for the arc length.

05:12 So as we do that, is n equal to from alpha to beta square root.

05:23 And so what is telling us to do is, okay, we have to square these equations.

05:29 So let me get cosine of theta minus.

05:40 This looks awful.

05:42 I completely understand, but this will probably go away with trig properties.

05:54 We'll probably have things even cancel out.

05:57 I'm not sure how this is going to play out, but we're going to see very shortly.

06:02 So here we go.

06:04 Here's a big old fun expression.

06:06 I'm actually just going to simply expand it.

06:11 It's completely fine.

06:15 All right, so a good way to do it.

06:19 So i'm going to have this times this, right? mabit times itself, right? because since i'm squaring, what i'm going to do is i'm going to be like, okay, hey, i'm going to square the first term.

06:31 So that's then going to give me square root.

06:35 I'll stop writing in black.

06:42 It's square root...

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