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 \(\boxed{41}\).