More limit trouble

[MarkSA]

Hello,

I'm still having trouble finding the exact limits to integrate from when finding areas enclosed by a polar curve.

In the problem:
r^2 = sin(2theta)

I basically just eyeballed the graph and plugged in values to find that a half the area of one of the two propellers is made from 0 to pi/4. Then I used lines of symmetry y = x and y = -x to say that the total area of the propellers is:
A = 4 * Integral from 0 to pi/4 of: 1/2 * [sin(2theta)] dtheta

In this problem, is there a better way to find the limits to integrate from? I'm worried that I will be in trouble if Iget an odd curve that I nee to use some strange limits to integrate, like 13pi/7.

 

[Deleted member 4993]

Hello,

I'm still having trouble finding the exact limits to integrate from when finding areas enclosed by a polar curve.

In the problem:
r^2 = sin(2theta)

Look at the given function.

r = min (= 0) when

\(\displaystyle \theta \, = \, 0\)

and

r = max (= 1) when

\(\displaystyle 2\cdot\theta \, = \, \frac{\pi}{2}\)

those are your limits.

I basically just eyeballed the graph and plugged in values to find that a half the area of one of the two propellers is made from 0 to pi/4. Then I used lines of symmetry y = x and y = -x to say that the total area of the propellers is:
A = 4 * Integral from 0 to pi/4 of: 1/2 * [sin(2theta)] dtheta

In this problem, is there a better way to find the limits to integrate from? I'm worried that I will be in trouble if Iget an odd curve that I nee to use some strange limits to integrate, like 13pi/7.