Answer
$62$
Work Step by Step
Step 1. Graph the function for the given interval of $-1\leq x\leq 8$ and identify a zero at $x=1$, which separates the positive and negative regions.
Step 2. The total area is the sum of the absolute value of the two separate areas $A_1$ and $A_2$.
Step 3. Use symmetry and evaluate
$A_1=2\int_0^1(5-5x^{2/3})dx=2(5x-3x^{5/3})|_0^1=2(5(1)-3(1)^{5/3})=4$
Step 4. Evaluate
$A_2=\int_1^8(5-5x^{2/3})dx=(5x-3x^{5/3})|_1^8=(5(8)-3(8)^{5/3})-(5(1)-3(1)^{5/3})=-58$
Step 5. Thus, we have the area as $A=|A_1|+|A_2|=62$
