题目

设曲线L的方程为y=1/4x^2-1/2lnx(1≤ x≤e)-|||- .（1）求L的弧长.（2）设D是由曲线L，直线x=1，x=e及x轴所围成的平面图形，求D的形心的横坐标.

设曲线L的方程为 $y%3D%5Cdfrac%7B1%7D%7B4%7Dx%5E%7B2%7D-%5Cdfrac%7B1%7D%7B2%7D%5Cln%20x%5Cleft%281%5Cleqslant%20x%5Cleqslant%20e%5Cright%29$ .

（1）求L的弧长.

（2）设D是由曲线L，直线x=1，x=e及x轴所围成的平面图形，求D的形心的横坐标.

题目解答

答案

$%5Cleft%28I%5Cright%29%5Cbecause%20ds%3D%5Csqrt%20%7B1%2By%27%5E%7B2%7D%7Ddx%3D%5Csqrt%20%7B1%2B%5Cleft%28%5Cdfrac%7B1%7D%7B2%7Dx-%5Cdfrac%7B1%7D%7B2x%7D%5Cright%29%5E%7B2%7D%7Ddx%3D%5Cdfrac%7B1%7D%7B2%7D%5Cleft%28x%2B%5Cdfrac%7B1%7D%7Bx%7D%5Cright%29dx$

$%5Ctherefore%20s%3D%5Cdfrac%7B1%7D%7B2%7D%5Cint_%7B1%7D%5E%7Be%7D%5Cleft%28x%2B%5Cdfrac%7B1%7D%7Bx%7D%5Cright%29%5Cmathrm%7Bd%7Dx%3D%5Cdfrac%7B1%7D%7B2%7D%5Cleft%5B%5Cdfrac%7B1%7D%7B2%7Dx%5E%7B2%7D%2B%5Cln%20x%5Cright%5D_%7B1%7D%5E%7Be%7D%3D%5Cdfrac%7Be%5E%7B2%7D%2B1%7D%7B4%7D$

$%5Cleft%28II%5Cright%29$ 设形心坐标为 $%28%5Coverline%7B%5Cdot%7Bx%7D%7D$ ， $%5Coverline%7B%5Cdot%7By%7D%7D%29$ ，又 $D%3D%5Cleft%28x%2Cy%5Cright%29%7C1%5Cleqslant%20x%5Cleqslant%20e$ ， $0%5Cleqslant%20y%5Cleqslant%20%5Cdfrac%7Bx%5E%7B2%7D%7D%7B4%7D-%5Cdfrac%7B%5Cln%20x%7D%7B2%7D$

则 $%5Coverline%7B%5Cdot%7Bx%7D%7D%3D%5Cdfrac%7B%E2%88%AC%20Dxdxdy%7D%7B%E2%88%AC%20Ddxdy%7D%3D%5Cdfrac%7B%5Cint_%7B1%7D%5E%7Be%7Dx%5Cmathrm%7Bd%7Dx%5Cint_%7B0%7D%5E%7B%5Cfrac%7Bx%5E%7B2%7D%7D%7B4%7D-%5Cfrac%7B%5Cln%20x%7D%7B2%7D%7D%5Cmathrm%7Bd%7Dy%7D%7B%5Cint_%7B1%7D%5E%7Be%7D%5Cmathrm%7Bd%7Dx%5Cint_%7B0%7D%5E%7B%5Cfrac%7Bx%5E%7B2%7D%7D%7B4%7D-%5Cfrac%7B%5Cln%20x%7D%7B2%7D%7D%5Cmathrm%7Bd%7Dy%7D%3D%5Cdfrac%7B%5Cint_%7B1%7D%5E%7Be%7Dx%5Cleft%28%5Cdfrac%7Bx%5E%7B2%7D%7D%7B4%7D-%5Cdfrac%7B%5Cln%20x%7D%7B2%7D%5Cright%29%5Cmathrm%7Bd%7Dx%7D%7B%5Cint_%7B1%7D%5E%7Be%7D%5Cleft%28%5Cdfrac%7Bx%5E%7B2%7D%7D%7B4%7D-%5Cln%20x%5Cright%29%5Cmathrm%7Bd%7Dx%7D%3D%5Cdfrac%7B%5Cleft%5B%5Cdfrac%7Bx%5E%7B4%7D%7D%7B16%7D-%5Cdfrac%7B1%7D%7B4%7Dx%5E%7B2%7D%5Cln%20x%2B%5Cdfrac%7B1%7D%7B8%7Dx%5E%7B2%7D%5Cright%5D_%7B1%7D%5E%7Be%7D%7D%7B%5Cleft%5B%5Cdfrac%7Bx%5E%7B3%7D%7D%7B12%7D-x%5Cln%20x%2Bx%5Cright%5D_%7B1%7D%5E%7Be%7D%7D%3D%5Cdfrac%7B3%5Cleft%28e%5E%7B4%7D-2e%5E%7B2%7D-3%5Cright%29%7D%7B4%5Cleft%28e%5E%7B3%7D-7%5Cright%29%7D$

设曲线L的方程为y=1/4x^2-1/2lnx(1≤ x≤e)-|||-__ __.（1）求L的弧长.（2）设D是由曲线L，直线x=1，x=e及x轴所围成的平面图形，求D的形心的横坐标.

题目解答

答案

设曲线L的方程为y=1/4x^2-1/2lnx(1≤ x≤e)-|||- .（1）求L的弧长.（2）设D是由曲线L，直线x=1，x=e及x轴所围成的平面图形，求D的形心的横坐标.