Find the area of a petal sketch.

**mymony1027** · March 15th 2011, 03:06 PM

Find the area of a petal sketch.
1) r= 4 sin(2theta)
2) r= 4sin(theta)
3) Compare and contrast conjecture.

Please please please help me with this calculus problem. I am so lost in my class. step by step work would be greatly appreciated. The work does not have to be completely accurate, I just need the steps. Thank you in advance.

**skeeter** · March 15th 2011, 05:31 PM

Originally Posted by mymony1027

Find the area of a petal sketch.
1) r= 4 sin(2theta)
2) r= 4sin(theta)
3) Compare and contrast conjecture.

Please please please help me with this calculus problem. I am so lost in my class. step by step work would be greatly appreciated. The work does not have to be completely accurate, I just need the steps. Thank you in advance.

help with #1 ... know how to sketch a polar graph? if not, you need to relearn how ... it's critical for determining area of a polar graph.

note that one petal gets formed for $r = 4\sin(2\theta)$ by values of $\theta$ , $0 \le \theta \le \dfrac{\pi}{2}$

using the form for area of polar curves ... $\displaystyle A = \int_{\theta_1}^{\theta_2} \frac{r^2}{2} \, d\theta$ ...

$\displaystyle A = \int_{0}^{\frac{\pi}{2}} 8\sin^2(2\theta) \, d\theta$

use the power reduction identity ...

$\sin^2(2\theta) = \dfrac{1 - \cos(4\theta)}{2}$ to help in the integration.

$\displaystyle A = 4\int_{0}^{\frac{\pi}{2}} 1 - \cos(4\theta) \, d\theta$

$4\left[\theta - \dfrac{\sin(4\theta)}{4}\right]_0^{\frac{\pi}{2}}$

$4\left[\dfrac{\pi}{2} - 0\right] = 2\pi$

o.k. you try #2.

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