设 $f(x)$ 为连续函数，则 $\int_0^{\frac{\pi}{4}} d \theta \int_{\frac{1}{\cos \theta+\sin \theta}}^{\sec \theta} f\left(r^2\right) 2 r d r=(\quad)$
A
$\int_0^1 d x \int_{1-x}^x f\left(x^2+y^2\right) d y$
B
$\int_0^1 d x \int_{1-x}^1 f\left(x^2+y^2\right) d y$
C
$\int_0^{\frac{1}{2}} d y \int_{1-y}^1 f\left(x^2+y^2\right) d x+\int_{\frac{1}{2}}^1 d y \int_y^1 f\left(x^2+y^2\right) d x$
D
$\int_0^1 d x \int_x^1 f\left(x^2+y^2\right) d y$
E
F