REPEATED INTEGRALS : #repeated #integrals
Repeated Integral :
Let f(x) be a bounded function defined on the rectangle R=[a,b;c,d].
Suppose ∫ f(x ,y) dy exists for each x Ꜫ [a ,b] and y limits from c to d. If g(x) = ∫ f( x ,y )dy, y limits from c to d , then g(x) is a function on [a ,b] . If g(x) is also integrable on [a ,b] , then ∫g(x)dx = ∫[ ∫ f(x , y)dy ] dx is called a repeated integral of f(x ,y) on R where x limits from a to b and y limits from c to d.
Problems on Repeated Integrals :
1. Evaluate ∫[ ∫ (x+y+1) dx ] dy and ∫[∫(x+y+1) dy]dx , x limits from -1 to 1 and y limits are from -1 to 0.
2. Show that for the function f(x ,y) defined on [ 0,1;0,1] with f( x ,y ) = 1/2, y is rational and f( x , y )= x, y is irrational, one of the repeated integral exists and the other does not exist.
3. Examine the nature of the repeated integral for the function f( x , y ) = 1/y^2 , if 0<x<y<1 and
f( x , y )= -1/x^2 , if 0<y<x<1
SOLUTIONS :
1.Evaluate ∫[ ∫ (x+y+1) dx ] dy and ∫[∫(x+y+1) dy]dx , x limits from -1 to 1 and y limits are from -1 to 0.
Sol :
2. Show that for the function f(x ,y) defined on [ 0,1;0,1] with f( x ,y ) = 1/2, y is rational and f( x , y )= x, y is irrational, one of the repeated integral exists and the other does not exist.
Sol :
3. Examine the nature of the repeated integral for the function f( x , y ) = 1/y^2 , if 0<x<y<1 and f( x , y )= -1/x^2 , if 0<y<x<1
Sol :
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#repeated #integrals
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