Definite Integrals Question 9

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

-2

0

2

-2

and

2

Approach 1

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.

Approach 2

\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}