( 842, #7) Calculate the iterated integral

Solution: The value of the double integral is 0.

Before integrating note that the domain of integration is symmetric with respect to the y-axis and f(x,y) is anti-symmetric with respect to the y-axis. That is,

Furthermore, since the integration on x is from -1 to 1 the value of the integral will be 0. The graph and evaluation bears this out:

f := (x,y)->x^3 * y^3 + 3*x*y^2:

c := x -> 0: d := x ->1:

Graph_dydx(f,-1,1,c,d, 9);

Integrate_dydx(f,-1,1,c,d);

f(x,y) = x^3*y^3+3*x*y^2 c(x) = 0 d(x) = 1

int(f(x,y), y) = 1/4*x^3*y^4+x*y^3

int(f(x,y), y = c(x) .. d(x)) = 1/4*x^3+x

int( int(f(x,y), y = c(x) ..d(x)), x) = 1/16*x^4+1/2*x^2

int(int(f(x,y), y = c(x) ..d(x)), x = a .. b) = 0

by: rjm