Answer to Question #188647 in Real Analysis for Nikhil

Question

The function f, defined by f(x,y)=x^3+xy+y, is inegtrable on [1,2]×[1,3]. True or false with full explanation.

Answer 1

The given function is-

$f(x,y)=x^3+xy+y$

Given region is $[1,2]×[1,3]={(1,1),(1,3),(2,1),(2,3)}$

Differentiate f w.r.t x-

$f_x(x,y)=3x^2+y$

Since (x,y) has positive value so value of $f_x(x,y)> 0.$

Differentiate f w.r.t y-

$f_y(x,y)=x+1$

Since x has the positive values so $f_y(x,y)>0.$

Differentiate $f_x$ w.r.t y-

$f_{xy}(x,y)=1$

So, value of $f_x(x,y),f_y(x,y), f_{xy}(x,y$ ) are not equal to zero, So given function is integrable.