Question

State the Fubini’s Theorem for the Triple Integrals and list all six possible order of

integration

Answer 1

$\text{Fubini's theorem for triple integral states that if f(x,y,z) is continuous on a rectangular}\\ \text{box D}=[a,b]\times[c,d]\times[e,f], \text{then}\\ \int\int\int_D\text{f(x,y,z) dV}=\int^f_e \int^d_c \int^b_a\text{f(x,y,z) dxdydz}.\\ \text{Furthermore, for a,b,c,d,e and f real numbers, the iterated triple integral can be}\\ \text{expressed in six different orderings:}\\ \int^f_e\int^d_c\int^b_a \text{f(x,y,z) dxdydz}=\int^f_e(\int^d_c(\int^b_a \text{f(x,y,z) dx)dy)dz}\\ =\int^d_c(\int^f_e(\int^b_a \text{f(x,y,z) dx)dz)dy}\\ =\int^b_a(\int^f_e(\int^d_c \text{f(x,y,z) dy)dz)dx}\\ =\int^f_e(\int^b_a(\int^d_c \text{f(x,y,z) dy)dx)dz}\\ =\int^d_c(\int^b_a(\int^f_e \text{f(x,y,z) dz)dx)dy}\\ =\int^b_a(\int^d_c(\int^f_e \text{f(x,y,z) dz)dy)dx}\\$