Mathematics, 07.04.2020 22:19 martymangaroo
1. Consider the following polar curve: r = 3 + 2 cos θ (a) Sketch the curve. (b) Find the area it encloses. (c) Set up an integral that represents the length of one loop of the curve. 2. Consider the following polar curve: r = 4 cos 3θ (a) Sketch the curve. (b) Find the area it encloses. (c) Set up an integral that represents the length of one loop of the curve.
Answers: 3
![A=\frac{1}{2}\int_\alpha^\beta [r(\theta)]^2d\theta\\\\A=\frac{1}{2}\int_{0}^{2\pi}[3+2cos\theta]^2d\theta\\\\A=\frac{1}{2}\int_{0}^{2\pi}[9+12cos\theta+4cos^2\theta]d\theta\\\\A=\frac{1}{2}[9(2\pi)+12sin(2\pi)+2(2\pi)+sin(4\pi)]\\\\A=11\pi](/tpl/images/0587/5727/9d158.png)
![A=6*\frac{1}{2}\int_{-\frac{\pi}{6}}^{\frac{pi}{6}}16cos^2\theta d\theta\\\\A=6[4\sqrt{3}+\frac{8\pi}{3}]=24\sqrt{3}+16\pi](/tpl/images/0587/5727/9be88.png)
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1. Consider the following polar curve: r = 3 + 2 cos θ (a) Sketch the curve. (b) Find the area it en...
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