Ushbu muammo uchun muammoning chegaralarini olish juda qiyin emas. Grafikning \(y\) o'qining chap tomonidagi qismini kuzatish uchun \(\frac{\pi }{2} \le \theta \le \frac{{3\pi }}{2}\) chegaralaridan foydalanishimiz kerak bo'ladi.

Esda tutingki, \(\theta \) ning kichikroq qiymatdan kattaroq qiymatga o'sishi sifatida bizni qiziqtirgan maydonni belgilovchi egri chiziq qismini kuzatish muhimdir.

Keyin hudud,

\[\begin{align*}A & = \int_{{\frac{\pi }{2}}}^{{\frac{{3\pi }}{2}}}{{\frac{1}{2}{{\left( {7 + 3\cos \theta } \right)}^2}\,d\theta }}\\ & = \frac{1}{2}\int_{{\frac{\pi }{2}}}^{{\frac{{3\pi }}{2}}}{{49 + 42\cos \theta + 9co{s^2}\left( \theta \right)\,d\theta }}\\ & = \frac{1}{2}\int_{{\frac{\pi }{2}}}^{{\frac{{3\pi }}{2}}}{{49 + 42\cos \theta + \frac{9}{2}\left( {1 + \cos \left( {2\theta } \right)} \right)\,d\theta }}\\ & = \frac{1}{2}\int_{{\frac{\pi }{2}}}^{{\frac{{3\pi }}{2}}}{{\frac{{107}}{2} + 42\cos \theta + \frac{9}{2}\cos \left( {2\theta } \right)\,d\theta }}\\ & = \left. {\frac{1}{2}\left( {\frac{{107}}{2}\theta + 42\sin \left( \theta \right) + \frac{9}{4}\sin \left( {2\theta } \right)} \right)} \right|_{\frac{\pi }{2}}^{\frac{{3\pi }}{2}} = \require{bbox} \bbox[2pt,border:1px solid black]{{42.0376}}\end{align*}\]

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.