Area of a Polar Curve Above the Polar Axis

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May 14th 2009, 12:36 PM
andyaddition

Area of a Polar Curve Above the Polar Axis

find the area of the interior of r=1-sin theta
(above the polar axis)

i think it is pi/4 but i also got pi/8 and i'm not sure which one, or if either, is right... thank you in advance to anyone who helps
May 14th 2009, 01:13 PM
galactus

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Is this what you're looking for?.

$\frac{1}{2}\int_{0}^{\pi}(1-sin(t))^{2}dt=\frac{3\pi-8}{4}$

This gives the area from 0 to Pi. In other words, the smaller portion above the 'x-axis', so to speak.
May 14th 2009, 01:22 PM
andyaddition

yeah. i did it from 0 to pi/2 though...why is it pi?
May 14th 2009, 01:33 PM
galactus

I just integrated from 0 to Pi. You can do it from 0 to Pi/2 and multiply by 2 as well to get the same thing.

The problem said area above the polar axis. It starts at what we can say is the positive x-axis and turns counterclockwise to Pi so that it encompasses all the area above the polar axis. After you pass Pi(180 degrees) you get below it. I shaded the wanted region, but it did not appear when I saved it to post. Sorry 'bout that.
May 14th 2009, 01:36 PM
Soroban

Hello, andyaddition!

Did you make a sketch?

Quote:

Find the area of the interior of $r\:=\:1-\sin\theta$ (above the polar axis)

This is a cardioid . . .

Code:

| ..*.. | ..*.. *:::::::* | *:::::::* - - * - - - - - * - - - - - * - - * | * | * | * * | * * | * | * | * * | * * | * * | * * * * |

We can find the area from $\theta = 0$ to $\theta = \tfrac{\pi}{2}$ . . . and double.

So we have:. $A \;=\;2 \times \tfrac{1}{2}\int^{\frac{\pi}{2}}_0(1 -\sin\theta)^2\,d\theta$

I get: . $\frac{3\pi}{4}-2 \:\approx\:0.3562$
May 14th 2009, 01:38 PM
galactus

Quote:

Originally Posted by Soroban

Hello, andyaddition!

Did you make a sketch?

This is a cardioid . . .

Code:

| ..*.. | ..*.. *:::::::* | *:::::::* - - * - - - - - * - - - - - * - - * | * | * | * * | * * | * | * | * * | * * | * * | * * * * |

We can find the area from $\theta = 0$ to $\theta = \tfrac{\pi}{2}$ . . . and double.

So we have:. $A \;=\;2 \times \tfrac{1}{2}\int^{\frac{\pi}{2}}_0(1 -\sin\theta)^2\,d\theta$

I get: . $\frac{3\pi}{4}-2 \:\approx\:0.3562$

There ya' go. I just did it from 0 to Pi. Gives the same thing.
May 14th 2009, 01:41 PM
andyaddition

thank you both very much