The double integral $$\int\limits_0^{\text{a}} {\int\limits_0^{\text{y}} {{\text{f}}\left( {{\text{x, y}}} \right){\text{dx dy}}} } $$ is equivalent to A. $$\int\limits_0^{\text{x}} {\int\limits_0^{\text{y}} {{\text{f}}\left( {{\text{x, y}}} \right){\text{dx dy}}} } $$ B. $$\int\limits_0^{\text{a}} {\int\limits_{\text{x}}^{\text{y}} {{\text{f}}\left( {{\text{x, y}}} \right){\text{dx dy}}} } $$ C. $$\int\limits_0^{\text{a}} {\int\limits_{\text{x}}^{\text{a}} {{\text{f}}\left( {{\text{x, y}}} \right){\text{dy dx}}} } $$ D. $$\int\limits_0^{\text{a}} {\int\limits_0^{\text{a}} {{\text{f}}\left( {{\text{x, y}}} \right){\text{dx dy}}} } $$

The correct answer is $\boxed{\text{(C)}}$.

The double integral $\int_0^a \int_0^y f(x, y) \, dx dy$ is the integral of the function $f(x, y)$ over the region in the $xy$-plane bounded by the lines $x = 0$, $x = a$, $y = 0$, and $y = a$. The order of integration does not matter in double integrals, so the integral is also equal to $\int_0^a \int_x^a f(x, y) \, dy dx$.

The other options are incorrect because they do not integrate over the same region. Option $\text{(A)}$ integrates over the region bounded by the lines $x = 0$, $x = y$, $y = 0$, and $y = a$. Option $\text{(B)}$ integrates over the region bounded by the lines $x = 0$, $x = a$, $y = x$, and $y = a$. Option $\text{(D)}$ integrates over the entire $xy$-plane.