Find the total area between the $$ x $$-axis and the function on the given interval. $$ f(x)=x^{\frac{1}{3}}-x $$ $$ [-1,8] $$ $$ [?] $$

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To find the total area between the $$ x $$-axis and the function $$ f(x) = x^{\frac{1}{3}} - x $$ on the interval $$[-1, 8]$$, we need to calculate the definite integral of the function over the given interval.

The definite integral of a function $$ f(x) $$ over an interval $$[a, b]$$ is given by:

$$ \int_{a}^{b} f(x) \, dx $$

In this case, the function is $$ f(x) = x^{\frac{1}{3}} - x $$ and the interval is $$[-1, 8]$$.

Let's calculate the definite integral:
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int_{-1}^{8} \left(x^{\frac{1}{3}}-x\right) dx$$
- step1: Evaluate the integral:
  $$\int \left(x^{\frac{1}{3}}-x\right) dx$$
- step2: Use properties of integrals:
  $$\int x^{\frac{1}{3}} dx+\int -x dx$$
- step3: Evaluate the integral:
  $$\frac{3}{4}x^{\frac{4}{3}}+\int -x dx$$
- step4: Evaluate the integral:
  $$\frac{3}{4}x^{\frac{4}{3}}-\frac{x^{2}}{2}$$
- step5: Return the limits:
  $$\left(\frac{3}{4}x^{\frac{4}{3}}-\frac{x^{2}}{2}\right)\bigg |_{-1}^{8}$$
- step6: Calculate the value:
  $$-\frac{81}{4}$$
The total area between the $$ x $$-axis and the function $$ f(x) = x^{\frac{1}{3}} - x $$ on the interval $$[-1, 8]$$ is approximately $$-20.25$$.
The total area between the $$ x $$-axis and the function $$ f(x) = x^{\frac{1}{3}} - x $$ on the interval $$[-1, 8]$$ is approximately $$-20.25$$.

Find the total area between the
-axis and the function on the
given interval.

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Find the total area between the -axis and the function on the given interval.