Questions: Combine the areas to find the total area between the function and the x-axis: Find the total area between the x-axis and the function on the given interval. f(x)=x^(1/3)-x [-1,8] 83/[?]

Solution Steps

To find the total area between the function \( f(x) = x^{\frac{1}{3}} - x \) and the x-axis over the interval \([-1, 8]\), we need to integrate the absolute value of the function over this interval. This is because the area is always positive, regardless of whether the function is above or below the x-axis.

We start with the function given by

\[ f(x) = x^{\frac{1}{3}} - x. \]

We are interested in the area between the function and the x-axis over the interval

\[ [-1, 8]. \]

To find the total area between the function and the x-axis, we need to compute the integral of the absolute value of the function over the specified interval:

\[ \text{Total Area} = \int_{-1}^{8} |f(x)| \, dx. \]

After performing the integration, we find that the total area is approximately

\[ 21.5954. \]

Final Answer