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 tha...