Find the area between the following curves.
\[
x=1, x=3, y=x^{3}-8, \text { and } y=0
\]
The area between the curves is $\square$.
(Simplify your answer.)
Answer
Final Answer: The area between the curves is \(\boxed{4}\).
Step 1 ：We are given the curves \(x=1\), \(x=3\), \(y=x^{3}-8\), and \(y=0\).
Step 2 ：We need to find the area between these curves.
Step 3 ：The area under the curve \(y = 0\) is just the x-axis, so we are essentially finding the area between the curve \(y = x^3 - 8\) and the x-axis from \(x = 1\) to \(x = 3\).
Step 4 ：The area is given by the definite integral of the function from \(x = 1\) to \(x = 3\).
Step 5 ：After calculating, we find that the area between the curves is 4.
Step 6 ：Final Answer: The area between the curves is \(\boxed{4}\).
