題目 4. (10 points) Find the arc length of the curve with equation x2/3+y2/3=1x^{2/3}+y^{2/3}=1x2/3+y2/3=1 within the region {(x,y)∣x≥0 and y≥0}\{(x,y)\mid x\ge 0 \text{ and } y\ge 0\}{(x,y)∣x≥0 and y≥0}. 解答 解法一 第一象限的一段星形線可參數化為 x=cos3tx=\cos^3tx=cos3t、y=sin3ty=\sin^3ty=sin3t，0≤t≤π/20\le t\le\pi/20≤t≤π/2。因此 (dxdt)2+(dydt)2=3sintcost.\sqrt{\left(\frac{\mathrm{d}x}{\mathrm{d}t}\right)^2+ \left(\frac{\mathrm{d}y}{\mathrm{d}t}\right)^2} =3\sin t\cos t.(dtdx)2+(dtdy)2=3sintcost. 故弧長為 ∫0π/23sintcost dt=32.\int_0^{\pi/2}3\sin t\cos t\,\mathrm{d}t=\boxed{\frac32}.∫0π/23sintcostdt=23.
