The entire length of the cardioid r=5(1+cosθ) is (a) 40 (b) 30 (c) 20 (d) 5.

Name: Problem 13, Chapter 6: Higher Engineering Mathematics
Uploaded: 2022-04-11T03:06:18-08:00
Duration: 8 min 53 s
Channel: Jeff Vermeire
Description: The entire length of the cardioid r=5(1+cosθ) is (a) 40 (b) 30 (c) 20 (d) 5.

step 1 We are given the equation of a cardioid and want to find its length. So the first thing I want t

The entire length of the cardioid r=5(1+cosθ) is (a) 40 (b)...

Key Concepts

Polar Coordinates

Polar coordinates provide a method of representing points in a plane using a distance from a reference point and an angle from a reference direction. This system is particularly effective for describing curves that exhibit radial symmetry, such as those given by equations of the form r = f(?), simplifying the analysis and computation of various properties.

Cardioid

A cardioid is a specific type of curve that has a heart-like shape and is typically expressed in polar form as r = a(1 + cos ?) or r = a(1 + sin ?). It features properties such as a cusp (a pointed end) and symmetry, making it a common example for exploring concepts in calculus and geometry related to polar curves.

Arc Length in Polar Coordinates

Calculating the arc length of a curve in polar coordinates involves integrating the square root of the sum of r(?) squared and the derivative of r(?) squared with respect to ?. The formula, L = ??(r² + (dr/d?)²)d?, is fundamental for determining the total length of curves defined in polar coordinates.

Transcript

00:01 We are given the equation of a cardioid and want to find its length.

00:06 So the first thing i want to do is rewrite our equation as 5 plus 5 cosine theta.

00:18 And i'm going to take its derivative d theta equals negative 5 sine theta.

00:29 That little common handy.

00:32 As you can see directly below it here i've written the formula for the length of a cardioid and we just have to plug in these numbers and chug.

00:45 So for the length of one cycle formula is specifically two times the integral from zero to pi times the square root of r squared which is five, five, theta squared plus negative 5 sine theta squared d theta.

01:24 I guess i'm going to carry out this multiplication to pi this integral is see 25 cosine squared theta plus 50 cosine theta plus 25 plus 25 x squared theta let's see one half power fee theta now if you look at this i can actually factor out these this 25 here since that's 5 squared that's going to give us 2 times 5 5 to scroll from 0 to pi of cosine squared theta plus 2, cosine theta plus 1 plus sine squared theta, 3 1⁄2, 0⁄2 power d theta, cosine squared data plus sine square theta equal 1.

03:02 So that helps us simplify it even more.

03:06 And this just becomes 10 times the integral square root 2, cosine, theta plus 1, which i can rewrite even one more time.

03:36 10, 0, pi, square root of 2 times square root of cosine of cosine, theta, theta, plus 1, d theta.

03:51 Now i have to use a tricking the metric identity.

03:57 A cosine of theta equal, oh, actually, excuse me, cosine of theta plus 1 equal 2, cosine squared theta over 2.

04:19 And i can plug that back in to our formula.

04:24 10 squared 2, 0 to pi, square root of 2, of 2 cosine squared theta over 2 d theta.

04:45 I can pull out the squared 2, since it's also a constant...

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