📝 题目
例 4 星形线的参数方程为
$$ x = a{\cos }^{3}t,\;y = a{\sin }^{3}t,\;0 \leq t \leq {2\pi }. $$
试求由星形线所围成的平面图形 $\Omega$ 的面积 (参看图 16-11).
\begin{center} ![](\"/static/img/math_analysis/092.jpg\")
\end{center} \hspace*{3em}
图 16-11
💡 答案与解析
解 我们有
$$ \sigma \left( \Omega \right) = \frac{1}{2}{\int }_{\mathrm{a}\Omega }x\mathrm{\;d}y - y\mathrm{\;d}x $$
$$ = \frac{1}{2}{\int }_{0}^{2\pi }\left\lbrack {a{\cos }^{3}t\left( {{3a}{\sin }^{2}t\cos t}\right) }\right. $$
$$ \left. {-a{\sin }^{3}t\left( {-{3a}{\cos }^{2}t\sin t}\right) }\right\rbrack \mathrm{d}t $$
$$ = \frac{3{a}^{2}}{2}{\int }_{0}^{2\pi }{\cos }^{2}t{\sin }^{2}t\mathrm{\;d}t $$
$$ = \frac{3{a}^{2}}{8}{\int }_{0}^{2\pi }{\sin }^{2}{2t}\mathrm{\;d}t $$
$$ = \frac{3{a}^{2}}{8}{\int }_{0}^{2\pi }\frac{1 - \cos {4t}}{2}\mathrm{\;d}t $$
$$ = \frac{3}{8}\pi {a}^{2}. $$