# Math Help - Is this me or the book?

1. ## Is this me or the book?

I am doing a question on finding the area enclosed by a loop, given by the equation:

$r = 2\cos^{\frac{3}{2}}\theta$ with $\frac{-\pi}{2} \le\theta\le \frac{\pi}{2}$.

I know that the area is $\frac{1}{2} \int r^2 d\theta$

Which works out to be $\frac{1}{2} \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} 4\cos^{3}\theta d\theta$

Taking out the 4 and expanding I end up with $2 \int_{\frac{-\pi}{2}}^{\frac{\pi}{2}} \cos{\theta}(1-\sin^{2}\theta) d\theta$

The book however has a 4 outside the brackets, is this a typo or something that I've missed?

Thanks for the help

2. I would have done the same as what you have; could the lower bound at the very last step have been changed in the book from -pi/2 to 0, which would then require to double the integral in order to get the full loop?

3. hey isnt it from -pi/2 to +pi/2 ??? how come it became -pi/2 to 1/pi ???
if its from -pi/2 to +pi/2 then u will have 4 outside... and integral will become from 0 to +pi/2....

4. Originally Posted by 000
hey isnt it from -pi/2 to +pi/2 ??? how come it became -pi/2 to 1/pi ???
if its from -pi/2 to +pi/2 then u will have 4 outside... and integral will become from 0 to +pi/2....
Sorry that was a typo on my part, the limits are $\frac{-\pi}{2} \le \theta \le \frac{\pi}{2}$.

Could you explain why it becomes 4 outside, sorry I don't quite understand?

Thanks for the replies

5. hey $\cos^3{\theta}$ is an even function... so integral from $-\pi/2$ to $+\pi/2$ becomes twice integral $0$ to $+\pi/2$

Since The integral of an even function from −A to +A is twice the integral from 0 to +A

hence u get four outside....

6. Of course do you!!! Does anyone else fail to see the blindingly obvious when doing complicated maths lol?

Thanks for the help

7. hehe... thats called carelessness.... I am a master at making these type of mistakes... hope i dont make any for tommorows university exam...