Chapter 14 - Multiple Integrals - 14.2 Double Integrals Over Nonrectangular Regions - Exercises Set 14.2 - Page 1016: 35

False.

The region is given by \[ \int_{-1}^{1} \int_{x^{2}}^{1} f(x, y) d y d x=\iint_{R} f(x, y) d A \] Let $x=f(x)$; we get \[ 0=\int_{-1}^{1} \int_{x^{2}}^{1} f(x, y) d y d x \] But \[ \frac{1}{4}=\int_{0}^{1} \int_{x^{2}}^{1} f(x, y) d y d x \] This is false in general cases. But if the function $f(x, y)$ is even in $x,$ it becomes true.