Answer
0
Work Step by Step
FTC: Let $f$ be a continuous function defined on the interval $[a, b]$,
and let $F$ be any antiderivative of f defined on $[a, b]$. Then
$\displaystyle \int_{a}^{b}f(x)dx=\left[F(x)\right]_{a}^{b}=F(b)-F(a)$.
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$f(x)=x^{1/3}$, which is continuous on [$-1,1$].
First, find an antiderivative $F(x)$:
$\displaystyle \int f(x)dx=\frac{x^{4/3}}{\frac{4}{3}}+C=\frac{3x^{4/3}}{4}+C$
(we take C=0, since by FTC we need ANY antiderivative)
$F(x)=\displaystyle \frac{3x^{4/3}}{4}$
Next, evaluate using: $(x^{4/3}=(x^{1/3})^{4}$, positive):
$F(1)=\displaystyle \frac{3(1)^{4}}{4}=\frac{3}{4}$
$F(-1)=\displaystyle \frac{3(-1)^{4}}{4}=\frac{3}{4}$
Finally, apply the theorem:
$\displaystyle \int_{-1}^{1}f(x)dx=F(1)-F(-1)$
$=\displaystyle \frac{3}{4}-\frac{3}{4}=0$
