Polar Area — Practice Problems and Worked Solutions

The formulas you need here

16 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Find the area enclosed by the polar curve.

   $$r=1,0\le \theta \le 2\pi$$

   Answer: $\pi$

   Step-by-step solution

   1. $$A=\frac{1}{2}{\int }_{\alpha }^{\beta }r(\theta {)}^{2}d\theta$$
   2. $$A=\frac{1}{2}{\int }_0^{2\pi }(1{)}^{2}d\theta =\pi$$
2. 2.

   Find the area enclosed by the polar curve.

   $$r=2,0\le \theta \le \pi$$

   Answer: $2\pi$

   Step-by-step solution

   1. $$A=\frac{1}{2}{\int }_{\alpha }^{\beta }r(\theta {)}^{2}d\theta$$
   2. $$A=\frac{1}{2}{\int }_0^{\pi }(2{)}^{2}d\theta =2\pi$$
3. 3.

   Find the area enclosed by the polar curve.

   $$r=\cos \left(\theta \right)+1,0\le \theta \le 2\pi$$

   Answer: $\frac{3\pi }{2}$

   Step-by-step solution

   1. $$A=\frac{1}{2}{\int }_{\alpha }^{\beta }r(\theta {)}^{2}d\theta$$
   2. $$A=\frac{1}{2}{\int }_0^{2\pi }(\cos \left(\theta \right)+1{)}^{2}d\theta =\frac{3\pi }{2}$$
4. 4.

   Find the area enclosed by the polar curve.

   $$r=2\sin \left(\theta \right),0\le \theta \le \pi$$

   Answer: $\pi$

   Step-by-step solution

   1. $$A=\frac{1}{2}{\int }_{\alpha }^{\beta }r(\theta {)}^{2}d\theta$$
   2. $$A=\frac{1}{2}{\int }_0^{\pi }(2\sin \left(\theta \right){)}^{2}d\theta =\pi$$
5. 5.

   Find the area enclosed by the polar curve.

   $$r=\sqrt{\cos \left(\theta \right)},0\le \theta \le \frac{\pi }{2}$$

   Answer: $\frac{1}{2}$

   Step-by-step solution

   1. $$A=\frac{1}{2}{\int }_{\alpha }^{\beta }r(\theta {)}^{2}d\theta$$
   2. $$A=\frac{1}{2}{\int }_0^{\frac{\pi }{2}}(\sqrt{\cos \left(\theta \right)}{)}^{2}d\theta =\frac{1}{2}$$
6. 6.

   Find the area enclosed by the polar curve.

   $$r=\sin \left(\theta \right)+1,0\le \theta \le \pi$$

   Answer: $2+\frac{3\pi }{4}$

   Step-by-step solution

   1. $$A=\frac{1}{2}{\int }_{\alpha }^{\beta }r(\theta {)}^{2}d\theta$$
   2. $$A=\frac{1}{2}{\int }_0^{\pi }(\sin \left(\theta \right)+1{)}^{2}d\theta =2+\frac{3\pi }{4}$$
7. 7.

   Find the area enclosed by the polar curve.

   $$r=3,0\le \theta \le 2\pi$$

   Answer: $9\pi$

   Step-by-step solution

   1. $$A=\frac{1}{2}{\int }_{\alpha }^{\beta }r(\theta {)}^{2}d\theta$$
   2. $$A=\frac{1}{2}{\int }_0^{2\pi }(3{)}^{2}d\theta =9\pi$$
8. 8.

   Find the area enclosed by the polar curve.

   $$r=4,0\le \theta \le \pi$$

   Answer: $8\pi$

   Step-by-step solution

   1. $$A=\frac{1}{2}{\int }_{\alpha }^{\beta }r(\theta {)}^{2}d\theta$$
   2. $$A=\frac{1}{2}{\int }_0^{\pi }(4{)}^{2}d\theta =8\pi$$
9. 9.

   Find the area enclosed by the polar curve.

   $$r=2,0\le \theta \le 2\pi$$

   Answer: $4\pi$

   Step-by-step solution

   1. $$A=\frac{1}{2}{\int }_{\alpha }^{\beta }r(\theta {)}^{2}d\theta$$
   2. $$A=\frac{1}{2}{\int }_0^{2\pi }(2{)}^{2}d\theta =4\pi$$
10. 10.

    Find the area enclosed by the polar curve.

    $$r=\cos \left(\theta \right),0\le \theta \le \frac{\pi }{2}$$

    Answer: $\frac{\pi }{8}$

    Step-by-step solution

    1. $$A=\frac{1}{2}{\int }_{\alpha }^{\beta }r(\theta {)}^{2}d\theta$$
    2. $$A=\frac{1}{2}{\int }_0^{\frac{\pi }{2}}(\cos \left(\theta \right){)}^{2}d\theta =\frac{\pi }{8}$$
11. 11.

    Find the area enclosed by the polar curve.

    $$r=\sin \left(\theta \right),0\le \theta \le \pi$$

    Answer: $\frac{\pi }{4}$

    Step-by-step solution

    1. $$A=\frac{1}{2}{\int }_{\alpha }^{\beta }r(\theta {)}^{2}d\theta$$
    2. $$A=\frac{1}{2}{\int }_0^{\pi }(\sin \left(\theta \right){)}^{2}d\theta =\frac{\pi }{4}$$
12. 12.

    Find the area enclosed by the polar curve.

    $$r=3\sin \left(\theta \right),0\le \theta \le \pi$$

    Answer: $\frac{9\pi }{4}$

    Step-by-step solution

    1. $$A=\frac{1}{2}{\int }_{\alpha }^{\beta }r(\theta {)}^{2}d\theta$$
    2. $$A=\frac{1}{2}{\int }_0^{\pi }(3\sin \left(\theta \right){)}^{2}d\theta =\frac{9\pi }{4}$$
13. 13.

    Find the area enclosed by the polar curve.

    $$r=2\cos \left(\theta \right),0\le \theta \le \frac{\pi }{2}$$

    Answer: $\frac{\pi }{2}$

    Step-by-step solution

    1. $$A=\frac{1}{2}{\int }_{\alpha }^{\beta }r(\theta {)}^{2}d\theta$$
    2. $$A=\frac{1}{2}{\int }_0^{\frac{\pi }{2}}(2\cos \left(\theta \right){)}^{2}d\theta =\frac{\pi }{2}$$
14. 14.

    Find the area enclosed by the polar curve.

    $$r=5,0\le \theta \le \frac{\pi }{2}$$

    Answer: $\frac{25\pi }{4}$

    Step-by-step solution

    1. $$A=\frac{1}{2}{\int }_{\alpha }^{\beta }r(\theta {)}^{2}d\theta$$
    2. $$A=\frac{1}{2}{\int }_0^{\frac{\pi }{2}}(5{)}^{2}d\theta =\frac{25\pi }{4}$$
15. 15.

    Find the area enclosed by the polar curve.

    $$r=\cos \left(\theta \right)+1,0\le \theta \le \pi$$

    Answer: $\frac{3\pi }{4}$

    Step-by-step solution

    1. $$A=\frac{1}{2}{\int }_{\alpha }^{\beta }r(\theta {)}^{2}d\theta$$
    2. $$A=\frac{1}{2}{\int }_0^{\pi }(\cos \left(\theta \right)+1{)}^{2}d\theta =\frac{3\pi }{4}$$
16. 16.

    Find the area enclosed by the polar curve.

    $$r=\sqrt{\sin \left(\theta \right)},0\le \theta \le \frac{\pi }{2}$$

    Answer: $\frac{1}{2}$

    Step-by-step solution

    1. $$A=\frac{1}{2}{\int }_{\alpha }^{\beta }r(\theta {)}^{2}d\theta$$
    2. $$A=\frac{1}{2}{\int }_0^{\frac{\pi }{2}}(\sqrt{\sin \left(\theta \right)}{)}^{2}d\theta =\frac{1}{2}$$

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