Question

4．已知曲線C₁：y²=2x與C₂：y=$\frac{1}{2}{x^2}$．求兩條曲線所圍圖形（如圖所示陰影部分）的面積S．

Answer 1

分析 根據(jù)定積分的幾何意義計(jì)算．

解答 解：解方程組$\left\{\begin{array}{l}{{y}^{2}=2x}\\{y=\frac{1}{2}{x}^{2}}\end{array}\right.$得$\left\{\begin{array}{l}{x=0}\\{y=0}\end{array}\right.$或$\left\{\begin{array}{l}{x=2}\\{y=2}\end{array}\right.$，
∴陰影面積S=${∫}_{0}^{2}$（$\sqrt{2x}$-$\frac{1}{2}{x}^{2}$）dx=（$\frac{2\sqrt{2}}{3}$x${\;}^{\frac{3}{2}}$-$\frac{1}{6}$x³）${|}_{0}^{2}$=$\frac{4}{3}$．

點(diǎn)評 本題考查了定積分在幾何中的應(yīng)用，屬于中檔題．