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샘플

10.3. 56

 

(c)

\(r=\cos \frac{\theta}{3}\)

[/pages/4176465/attachments/2110527 MSP74719784071iad69b2900005cid5ibigi61dg94.gif]

 http://www.wolframalpha.com/input/?i=r%3Dcos+(theta/3)

 

(d)

\(r=1+2\cos\theta\)

  [/pages/4176465/attachments/2110537 MSP11271978443hf7aa7h2f00002bd5bh2db919dc1f.gif] 

 http://www.wolframalpha.com/input/?i=r%3D1%2B2+cos+(theta)

(e) 

\(r=2+\sin 3\theta\)

http://www.wolframalpha.com/input/?i=r%3D2%2Bsin(3thetA)

 [/pages/4176465/attachments/2110539 MSP47019784919haafi5hc00002h164hgd33f4i46i.gif]

(f)

\(r=1+2\sin 3\theta\)

 http://www.wolframalpha.com/input/?i=r%3D1%2B2sin(3theta)

[/pages/4176465/attachments/2110535 MSP6981978475e4574h83h00003a7ga0a9g5d06c13.gif]

 

 

 

10.4.44.

MSP26619784bhf76f4e70900004e38231fbie4a798.gif

Note that \(2\sin^2 \alpha +2\sin \alpha-1=0\).

Since \(0< \alpha <\frac{\pi}{2}\), \(\sin \alpha =\frac{\sqrt 3 -1}{2}\) and \(\cos \alpha =\sqrt{\frac{\sqrt 3}{2}}\)

 

\(\frac{1}{2}\int_{\alpha}^{\pi-\alpha}r^2 d\theta=\int_{\alpha}^{\pi/2}r^2 d\theta=\int_{\alpha}^{\pi/2}64(1+\sin\theta)^2d\theta\)

\(=64\int_{\alpha}^{\pi/2}(1+\sin\theta)^2d\theta=64\int_{\alpha}^{\pi/2}1+2\sin\theta+\sin^2\theta d\theta\)

\(=64[\frac{3}{2}\theta - 2 \cos \theta- \frac{1}{4}\sin 2\theta]_{\alpha}^{\pi/2}=[96\theta - 128 \cos \theta- 16\sin 2\theta]_{\alpha}^{\pi/2}\)

\(=[96\theta - 128 \cos \theta- 16\sin 2\theta]_{\alpha}^{\pi/2}=48\pi-96\alpha+128\cos \alpha+16\sin 2\alpha\)

We need to subtract from this the area of the triangle which is given by \[2\times \frac{1}{2}\cdot 4 (8+8\sin\alpha)\sin (\pi/2-\alpha)=32(1+\sin\alpha)\cos \alpha=32\cos \alpha+16 \sin 2\alpha\]

So the area will be given by \[48\pi-96\alpha+128\cos \alpha+16\sin 2\alpha-(32\cos \alpha+16 \sin 2\alpha)=48\pi-96\alpha+96\cos \alpha\approx 204.16\]

http://www.wolframalpha.com/input/?i=N[48pi+-96(arcsin{(sqrt+3+-+1)/2})%2B96(sqrt((sqrt+3)/2)),10]

 

10.4.47.

\(r=\theta^2\)

\(L=\int_0^{2\pi}\sqrt{\theta^4+(2\theta)^2}\,d\theta=\int_0^{2\pi}\theta\sqrt{\theta^2+4}\,d\theta\)

 

 

10.4.55.

(b)

\(r^2=\cos 2\theta\)

[/pages/4176465/attachments/2110541 MSP71919784049c09bb98700001095e8fadhf5gc4d.gif]

http://www.wolframalpha.com/input/?i=r^2%3Dcos+(2theta)

 

 

encyclopedia

• http://www.wolframalpha.com/input/?i=r^2
  
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• http://www.wolframalpha.com/input/?i=
  
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