What are all values of \(k\) for which \( \displaystyle \int\limits_{-2}^k x^2   dx=0 \) ?

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The area under the curve will equal zero if the lower and upper limits of integration are equal.

$$ \int\limits_{-2}^{-2} x^2   dx=0 $$

Since the graph of \(x^2\) is always positive, taking any other values for the upper limit of integration would result in a nonzero value for the definite integral.

$$ \int\limits_{-2}^k x^2   dx=0 $$ $$ \frac{1}{3}x^3 \Big|_{-2}^k=0 $$ $$ = \frac{1}{3}k^3 - \frac{1}{3}(-2)^3 = 0 $$ $$ k^3 = -8 $$ $$ k = \boxed{-2} $$