含参常义积分和含参广义积分习题课教案 任课教师:赵萃魁辅导教师:贺飞 第十七章含参常义积分 基础知识 1.关于I()=f(x,y)dr (1)I(y)在可上连续的充分条件是∫(x,y)在[a,b;c,d上连续 (2)()在[c,小上可微且r(y)=mf(x,yd的充分条件是fr,y,f(x,y) 在[ab;c,④上连续 2.关于F(y) f(a, y)d (1)F(y)在[c,d上连续的充分条件是f(x,y)在[a,b;c,d上连续, a(y),b(y)在[cd上连续且a≤a(y)≤b,a≤b(y)≤b (2)F()在小上可微且F()=MmD(x,)dx+f(0),列b(o) fa(y),ya(y)的充分条件是f(x,y),f2(x,y)在a,b;c,可上连续,同时在 cd上b(y),a(y)存在且a≤a(y)≤b,a≤b(y)≤b. 3.关于积分可交换 dymf(x,y)dx=mdmf(x,y)d的充分条件是f(x,y)在bcd 上连续。 二、本章课后习题 求F(y) 解:因为e-y在RxR上连续,所以 F()=Jm(-2)-dx+c-n,2y-c-,1 2ye-y-ey- 2.F(y)=0(x+y)f(x)dx,其中f(x)是可微的,求F"(y)￾
￾
  ￾
       1.  I(y) =
b a f(x, y)dx. (1)I(y) Æ [c, d]   f(x, y) Æ [a, b; c, d]   (2)I(y) Æ [c, d]  I
(y) =
b a fy(x, y)dx  f(x, y), fy(x, y) Æ [a, b; c, d]   2.  F(y) =
b(y) a(y) f(x, y)dx. (1)F(y) Æ [c, d]   f(x, y) Æ [a, b; c, d]   a(y), b(y) Æ [c, d]   a
a(y)
b, a
b(y)
b. (2)F(y) Æ [c, d]  F
(y) =
b(y) a(y) fy(x, y)dx + f[b(y), y]b
(y) − f[a(y), y]a
(y)  f(x, y), fy(x, y) Æ [a, b; c, d]  Æ [c, d]  b
(y), a
(y) Æ a
a(y)
b, a
b(y)
b. 3. 
d c dy
b a f(x, y)dx =
b a dx
d c f(x, y)dy  f(x, y) Æ [a, b; c, d]    !"#$% 1. F(y) =
y2 y e−x2ydx, F
(y). !"&# e−x2y Æ R × R  $' F
(y) =
y2 y (−x2)e−x2ydx + e−y5 · 2y − e−y3 · 1 = 2ye−y5 − e−y3 −
y2 y x2e−x2ydx. 2. F(y) =
y 0 (x + y)f(x)dx, %( f(x)  F

(y). 1