題目 7. (10%) Find the area of the region bounded by the polar curve r=32+sinθr=\dfrac{3}{2+\sin\theta}r=2+sinθ3. Hint: sketch the curve first. 解答 解法一 A=12∫02π(32+sinθ)2 dθ.A=\frac12\int_0^{2\pi}\left(\frac3{2+\sin\theta}\right)^2\,\mathrm{d}\theta.A=21∫02π(2+sinθ3)2dθ. 使用 ∫02πdθ(a+bsinθ)2=2πa(a2−b2)3/2\displaystyle\int_0^{2\pi}\frac{\mathrm{d}\theta}{(a+b\sin\theta)^2}=\frac{2\pi a}{(a^2-b^2)^{3/2}}∫02π(a+bsinθ)2dθ=(a2−b2)3/22πa，得到 A=2π3.\boxed{A=2\pi\sqrt3}.A=2π3.
