高數打卡04

$1.求心臟線r=a(1+cos\theta)(a>0,0\leqslant\theta \leqslant2\pi)所圍區域之面積.$
$圖形：$

$\begin{aligned} &S=\int_{0}^{2 \pi} d \theta \int_{0}^{a(1+\cos \theta)} r d r \\ &=\frac{1}{2} \int_{0}^{2 \pi} a^{2}(1+\cos \theta)^{2} d \theta \\ &=(\frac{1}{2} a^{2} \theta+a^{2} \sin \theta+\frac{1}{4} a^{2} \theta+\frac{1}{8} a^{2} \sin 2\theta)|_{0} ^{2 \pi} \\ &=\frac{3}{2} \pi a^{2} \end{aligned}$
$2.求雙紐線r^2=4cos2\theta所圍區域的面積.$
$圖形：$

$\begin{aligned} &S=4 \int_{0}^{\frac{\pi}{4}} d \theta \int_{0}^{2 \sqrt{\cos 2 \theta}} r d r\\ &=4 \int_{0}^{\frac{\pi}{4}} 2 \cos 2 \theta d \theta\\ &=4 \end{aligned}$
$3.求心臟線r=1+sin\theta(0\leqslant\theta \leqslant2\pi)所圍區域中位於第一象限部分的面積.$
$圖形：$

$\begin{aligned} &S=\int_{0}^{\frac{\pi}{2}} d \theta \int_{0}^{1+\sin \theta} r d r\\ &=\frac{1}{2} \int_{0}^{\frac{\pi}{2}}\left(1+\sin \theta+\sin ^{2} \theta\right) d \theta\\ &=1+\frac{3}{8} \pi \end{aligned}$