Determine the surface of a cardioid

[member 587159]

Homework Statement

Consider the cardioid given by the equations:

$x = a(2\cos{t} - \cos{2t})$
$y = a(2\sin{t} - \sin{2t})$

I have to find the surface that the cardioid circumscribes, however, I don't know what limits for $t$ I have to take to integrate over. How can I know that, as I don't know how this shape looks like (or more precisely where it is located)?

Homework Equations

Integration formulas

The Attempt at a Solution

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I know how I have to solve the problem once I have the integral bounds, but I don't know how I have to determine these. In similar problems, we could always eliminate cost and sint by using the identity $cos^2 x + sin^2 x = 1$ but neither this nor another way to eliminate the cos, sin seems to work. This makes me think, would it be sufficient if I find the maximum and minimum x-coordinate in function of t using derivatives? Then I would have bounds to integrate over, but this seems like a lot of work.

 

[BvU]

How can I know that, as I don't know how this shape looks like

That's easy ! calculate a few points and connect the dots ! And when things start to repeat, you know you've passed the bound for $t$

 

[member 587159]

That's easy ! calculate a few points and connect the dots ! And when things start to repeat, you know you've passed the bound for $t$

I have to know what the bound exactly is, so I can use it as my integration bounds.

 

[BvU]

O ?

I have to know what the bound exactly is

But you already have an exact expression for the bound

 

[member 587159]

O ?
But you already have an exact expression for the bound

I don't think I understand what you mean. Can you elaborate?

 

[BvU]

You wrote down the equations in post #1. They exactly define the bound. A bit corny of me, sorry.
Did you make the sketch? Find the domain of $t$ ?

What would your integral look like ?

The section in your textbook/curriculum where this exercise appears: what's it about ?

 

[member 587159]

You wrote down the equations in post #1. They exactly define the bound. A bit corny of me, sorry.
Did you make the sketch? Find the domain of $t$ ?

What would your integral look like ?

The section in your textbook/curriculum where this exercise appears: what's it about ?

It's about integration: definite and indefinite integrals. The next chapter deals with arc length.

 

[BvU]

That answers the last of my three questions. What about the other two ?
let me add another one: do you know two ways to integrate to get the area of a circle ?

I am a bit evasive because I am convinced you will say 'of course' once you have found your way out of this on your own steam...

 

[member 587159]

That answers the last of my three questions. What about the other two ?
let me add another one: do you know two ways to integrate to get the area of a circle ?

I am a bit evasive because I am convinced you will say 'of course' once you have found your way out of this on your own steam...

Yes I can find the integral of $\sqrt{r^2 - x^2}$ so that wouldn't be the problem. It's late now but I will try the problem tomorrow again and answer your questions then.

 

[BvU]

That's one way. There's a quicker way too. Same way works for the cardioid (but it's admittedly less simple than for a circle -- still an easy integral).

First three (not two) questions in post # 6 are still open ...

And it's ruddy late here too..

 

[member 587159]

That's one way. There's a quicker way too. Same way works for the cardioid (but it's admittedly less simple than for a circle -- still an easy integral).

First three (not two) questions in post # 6 are still open ...

And it's ruddy late here too..

Yes I believe you can do x = cos t, y = sin t and then integrate over the correct bounds. I will answer the other questions tomorrow.

 

[BvU]

get a good sleep ?

x = cos t, y = sin t

Isn't good for an area (only one integrand), but it's in the right direction. You need a factor r in both.

$\sqrt{r^2 - x^2}\$ are vertical strips, and I'm trying to lure you towards investigating pie pieces (sectors, so to say). As you picked up correctly.

What would the integral for the cardioid area look like with this approach ?

Did you make the sketch? Find the domain of $t$ ?

 

[member 587159]

get a good sleep ?
Isn't good for an area (only one integrand), but it's in the right direction. You need a factor r in both.

$\sqrt{r^2 - x^2}\$ are vertical strips, and I'm trying to lure you towards investigating pie pieces (sectors, so to say). As you picked up correctly.

What would the integral for the cardioid area look like with this approach ?

I figured out how to solve the problem using your hints. Thanks a lot.

 

[BvU]

You're welcome.