Answer
$\displaystyle \frac{13}{30}$
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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$f(x)=4.3x^{2}-1$, which is continuous on [$-1,0$].
First, find an antiderivative $F(x)$:
$\displaystyle \int f(x)dx=\frac{4.3x^{3}}{3}-x+C$
(we take C=0, since by FTC we need ANY antiderivative)
$F(x)=\displaystyle \frac{4.3x^{3}}{3}-x$
Next, evaluate:
$F(0)=0$
$F(-1)=\displaystyle \frac{4.3(-1)}{3}-(-1)=1-\frac{43}{30}=-\frac{13}{30}$
Finally, apply the theorem:
$\displaystyle \int_{-1}^{0}f(x)dx=F(0)-F(-1)=$
$=0-(-\displaystyle \frac{13}{30})=\frac{13}{30}$
