Converting double integral to polar coordinates

[Geekchick]

Homework Statement

\int\int(rsin2\vartheta)drd\vartheta
sorry i don't see how to put the bounds in but they are 0<\vartheta<\pi/2 and 0<r<2acos\vartheta

Homework Equations

I know that r=sin\vartheta

The Attempt at a Solution

Im really not sure where to start my text is terrible. I really don't know how to treat it with the double angle.

thanks for any help

sorry meant to put converting polar to rectangular.

 

[tiny-tim]

Hi Geekchick!

(type \int_0^{\pi /2}\int_0^{2acos\theta} )

Hint: use sin2θ = 2sinθcosθ

 

[jmb]

I know that r=sin\vartheta

I don't think you mean that... (!)

Im really not sure where to start my text is terrible.

Well there are basically three things you need to do:

• Sketch the area of integration and try and work out how you would describe it in rectangular rather than polar coordinates.
• Convert between integrating infinitesimal units of polar area to infinitesimal units of rectangular area (you can do this either geometrically or mathematically).
• Convert the integrand into rectangular coordinates --- tiny-tim has already given you a hint on this.

Apologies if you already knew this! Post your attempts to these issues if tiny-tim's hint isn't enough and somebody will give you additional hints...

 

[Geekchick]

Can I work backwards from the definition \intf(x,y)dxdy = \intf(rcos\vartheta,rsin\vartheta)rdrd\vartheta and then I can say rdrd\vartheta=dA and so all I have is \int\intsinydxdy? And I'm terrible at bounds so where do I start there?

 

[Mark44]

What exactly is the problem? Is it to evaluate this integral?
\int_{\theta = 0}^{\pi /2} \int_{r = 0}^{2a cos \theta} cos(2 \theta) r dr d\theta
If that's the problem, then all you need to do is evaluate the integral.

If you're trying to convert this to an iterated integral in rectangular (Cartesian) coordinates, which is not at all obvious from the information you've given, then yes dA = dx dy = r dr d\theta, and x = r cos \theta, y = r sin \theta.

The region over which you're integrating is a half-circle of radius 2a, with center at (a, 0) in Cartesian coordinates. Keep in mind, though, that your integral is not the area of this region.

 

[Geekchick]

sin instead of cos but yes. my instructions are to convert to Cartesian and then evaluate. But I'm not sure what the integral will be and with what bounds once I convert it...evaluating should be easy.

 

[Mark44]

The region over which you're integrating is a half-circle of radius 2a, with center at (a, 0) in Cartesian coordinates.

This half-circle determines the limits of integration, whether as a polar iterated integral or a Cartesian iterated integral. Can you find its equation in terms of Cartesian coordinates?
Also, can you convert sin(2\theta) to Cartesian form?

These are what you need to do to get the Cartesian integral.

 

[jmb]

Can I work backwards from the definition \intf(x,y)dxdy = \intf(rcos\vartheta,rsin\vartheta)rdrd\vartheta and then I can say rdrd\vartheta=dA?

While this is right, I would really really suggest you don't just treat this as a definition but try to understand where it has come from. For a start this will mean it's not something you need to memorise but can just work it out if you ever forget, but more importantly it will help you to understand geometrically what is happening in integrals like this.

What I'm basically asking is, do you understand why it should be r\;{\rm d}r\;{\rm d}\theta and not just {\rm d}r\;{\rm d}\theta?

And I'm terrible at bounds so where do I start there?

Another poster has already told you what the bounds mean geometrically, but again I would strongly encourage you to try and sketch it out and make sure you can see this for yourself (of course you should always verify it mathematically after making an initial sketch).

Once you have a sketch, you need to find a way to parametrise the area of integration in Cartesian coordinates. To understand this it may be worthwhile looking through some simpler examples first. A good way to start to get a feel for the way you parametrise areas in integral bounds is to do some exercises in just changing the order of integration with Cartesian double integrals. Once you are happy with this then move on to simple Cartesian<->Polar transformations and get a good feel for that too. I've attached a few examples. Why don't you work through them and then see if this has taught you enough to solve the bounds problem here. If not we can provide additional hints...