设 $D=\left\{(x, y) \left\lvert\, x^{\frac{2}{3}}+y^{\frac{2}{3}} \leqslant t^{\frac{2}{3}}\right.\right\}$, 且 $f(x, y)=\left\{\begin{array}{ll}\frac{1-\cos ^3 \sqrt{x^2+y^2}}{x^2+y^2}, & (x, y) \neq(0,0), \\ a, & (x, y)=(0,0)\end{array}\right.$ 在
全平面连续.
(1) 求 $a$ 的值;
(2) 计算 $\lim _{t \rightarrow 0^{+}} \frac{1}{t^2} \iint_D f(x, y) \mathrm{d} x \mathrm{~d} y$.