Find the area under the curve $f(x)=4 x^{3}$ over the interval $[0,4]$. \[ \int_{0}^{4}\left(4 x^{3}\right) d x=\square \]

Step 1 ：Given the function \(f(x)=4x^3\), we are asked to find the area under the curve over the interval \([0,4]\).

Step 2 ：To find the area under the curve, we need to integrate the function over the given interval.

Step 3 ：The integral of \(f(x)=4x^3\) is \(F(x) = x^4\).

Step 4 ：We evaluate this integral from 0 to 4, which gives us \(F(4) - F(0)\).

Step 5 ：Substituting the values, we get \(4^4 - 0^4 = 256\).

Step 6 ：Final Answer: The area under the curve \(f(x)=4 x^{3}\) over the interval \([0,4]\) is \(\boxed{256}\).