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think my teacher might be wrong on something, need to check here. https://www.desmos.com/calculator/mdavtmnba2 for that graph, find the area inside the rose, but outside the circle.
teachers solutions say just integrate (4cos(2theta))^2-2^2 from 0 to pi/6 then use symmetry.
teacher used this equation here http://tutorial.math.lamar.edu/Classes/CalcII/PolarArea_files/eq0008MP.gif but im not sure if u can do that in this case?

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Other answers:

|dw:1438082191447:dw|
You want to find that area and multiply by 8 is it
ya
|dw:1438082672639:dw| but wouldnt that integral cut off there
Oh you're right, my bad, lets try again
\(\large \int\limits_{0}^{\pi/4}\frac{1}{2}(4\cos2\theta)^2~d\theta\) gives below shaded area |dw:1438082959464:dw|
\(\large \int\limits_{0}^{\pi/6}\frac{1}{2}(2)^2~d\theta\) gives below shaded area |dw:1438083157606:dw|
next one would be pi/6 to pi/4 for the cos(2theta)
\(\large \int\limits_{\pi/6}^{\pi/4}\frac{1}{2}(4\cos2\theta)^2~d\theta\) gives below shaded area |dw:1438083357364:dw|
does that look okay now ? :)
yep.
lets work this and see if it matches with your teacher's answer
just want you notice that we're adding and subtracting that last orange piece, which is simply a waste of work... evaluate the integrals and you will see what i mean
your method : http://www.wolframalpha.com/input/?i=%28%5Cint%5Climits_0%5E%7B%5Cpi%2F4%7D+1%2F2%284%5Ccos%282%5Ctheta%29%29%5E2+d%5Ctheta%29+-+%28+%5Cint%5Climits_0%5E%7B%5Cpi%2F6%7D+1%2F2*%282%29%5E2+d%5Ctheta%29+-+%28%5Cint%5Climits_%7Bpi%2F6%7D%5E%7B%5Cpi%2F4%7D+1%2F2%284%5Ccos%282%5Ctheta%29%29%5E2+d%5Ctheta%29 your teacher's method : http://www.wolframalpha.com/input/?i=%28%5Cint%5Climits_0%5E%7B%5Cpi%2F6%7D+1%2F2%284%5Ccos%282%5Ctheta%29%29%5E2+d%5Ctheta%29+-+%28+%5Cint%5Climits_0%5E%7B%5Cpi%2F6%7D+1%2F2*%282%29%5E2+d%5Ctheta%29
\(\large \int\limits_{0}^{\pi/\color{red}{6}}\frac{1}{2}(4\cos2\theta)^2~d\theta\) gives below shaded area |dw:1438084278198:dw|
dang, was hoping to get extra credit for proving it wrong.
Haha! you had the right idea, pretty sure you will find other opportunities to catch ur teacher's mistakes ;)

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