Question

高等数学，关于定积分的题目

高等数学，关于定积分的题目题目如图，求详细步骤，满意必采纳，谢谢

Answer 1

画图，求斜率，写方程，联立方程求交点。写出定积分表达式，计算定积分。

Answer 2

y²=2px
对x求导：2yy'=2p
y'=p/y
法线斜率：k=-y/p
(p/2,p)处的法线斜率：k=-p/p=-1
(p/2,p)处的法线方程：y-p=-1(x-p/2)
x=3p/2-y①
抛物线方程：x=y²/(2p)②
联解①②，得交点：A(p/2,p)、B(9p/2,-3p)
面积：S=∫(-3p,p)[(3p/2-y)-y²/(2p)]dy
=∫(-3p,p)[3p/2-y-y²/(2p)]dy
=[(3p/2)y-(1/2)y²-p³/(6p)]|(-3p/2,p)
=(3p/2)(p+3p/2)-(1/2)[p²-(-3p/2)²]-[p³-(-3p/2)³]/(6p)
=15p²/4+5p²/8-35p²/48
=180p²/48+30p²/48-35p²/48
=175p²/48

Answer 3

Answer 4

Answer 5

如图所示。