Use the definite integral to find the area between the $x$-axis and $f(x)$ over the indicated interval Check first to see if the graph crosses the $x$-ax
\[
f(x)=2 x^{3} ;[-3,1]
\]

The area between the $x$-axis and $f(x)$ is $\square$ (Simplify your answer.)

Step 1 ：The function \(f(x) = 2x^3\) crosses the x-axis at \(x=0\).

Step 2 ：We split the integral into two parts: one from \(-3\) to \(0\) and the other from \(0\) to \(1\).

Step 3 ：We calculate the definite integral of \(f(x)\) from \(-3\) to \(0\), which gives us \(-\frac{81}{2}\).

Step 4 ：We calculate the definite integral of \(f(x)\) from \(0\) to \(1\), which gives us \(\frac{1}{2}\).

Step 5 ：We take the absolute value of each integral to ensure we are calculating the area, which is always positive.

Step 6 ：We add the absolute values of the two areas together to get the total area, which is \(41\).

Step 7 ：Final Answer: The area between the x-axis and \(f(x)\) over the interval \([-3,1]\) is \(\boxed{41}\).