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

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Answer

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

Steps

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 $\boxed{256}$.