Endi biz \(\theta \) ning grafiklar kesishgan joydagi qiymatlarini aniqlashimiz kerak (oldingi bosqichda grafikdagi qora chiziqlar bilan ko'rsatilgan).

Ularni topish juda oson. Grafiklar kesishgan joyda ular \(r\) qiymati bilan bir xil bo'lishi kerakligini bilamiz. Shunday qilib,

\[\begin{align*}3 + 3\sin \theta & = 2\\ \sin \theta & = - \frac{1}{3}\hspace{0.5in} \Rightarrow \hspace{0.5in}\theta = {\sin ^{ - 1}}\left( { - \frac{1}{3}} \right) = - 0.3398\end{align*}\]

Bu grafiklar kesishgan to'rtinchi kvadrantdagi burchak.

Birlik aylanasining qisqacha chizmasidan ikkita grafik kesishgan uchinchi kvadrantdagi burchakni tezda olishimiz mumkin.

\[\theta = \pi + 0.3398 = 3.4814\]

Ushbu muammo uchun \(\theta \) diapazonlari \( - 0.3398 \le \theta \le 3.4814\) bo'ladi.

Grafikdan biz \(r = 3 + 3\sin \theta \) ushbu mintaqa uchun "tashqi" grafik ekanligini va \(r = 2\) "ichki" grafik ekanligini ko'rishimiz mumkin.

Keyin hudud,

\[\begin{align*}A & = \int_{{ - 0.3398}}^{{3.4814}}{{\frac{1}{2}\left[ {{{\left( {3 + 3\sin \theta } \right)}^2} - {{\left( 2 \right)}^2}} \right]\,d\theta }}\\ & = \frac{1}{2}\int_{{ - 0.3398}}^{{3.4814}}{{5 + 18\sin \theta + 9{{\sin }^2}\left( \theta \right)\,d\theta }}\\ & = \frac{1}{2}\int_{{ - 0.3398}}^{{3.4814}}{{5 + 18\sin \theta + \frac{9}{2}\left( {1 - \cos \left( {2\theta } \right)} \right)\,d\theta }}\\ & = \frac{1}{2}\int_{{ - 0.3398}}^{{3.4814}}{{\frac{{19}}{2} + 18\sin \theta - \frac{9}{2}\cos \left( {2\theta } \right)\,d\theta }}\\ & = \left. {\frac{1}{2}\left( {\frac{{19}}{2}\theta - 18\cos \left( \theta \right) - \frac{9}{4}\sin \left( {2\theta } \right)} \right)} \right|_{ - 0.3398}^{3.4814} = \require{bbox} \bbox[2pt,border:1px solid black]{{33.7074}}\end{align*}\]

Ushbu integrallarni bajarish uchun zarur bo'lgan trig manipulyatsiyalarini bajarishingiz mumkinligiga ishonch hosil qiling. Ushbu bo'limdagi integrallarning aksariyati bunday manipulyatsiyani o'z ichiga oladi. Agar ularni qanday qilishni eslamasangiz, orqaga qayting va ko'rib chiqing Trig funksiyalarni o'z ichiga olgan integrallar bo'limi.