Question

Definite integrals: Use the Fundamental Theorem of Calculus to find the exact values of each of the definite integrals that follow. Sketch the areas described by these definite integrals to determine whether your answers are reasonable.

${\int }_{-2}^2\left(x^2-1\right)dx$

Short answer

The solution is: $\frac{4}{3}$

Step-by-step solution

Given

The given integral is: ${\int }_{-2}^2\left(x^2-1\right)dx$

To Find

Use the Fundamental Theorem of Calculus to find the exact values of each of the definite integrals that follow. Sketch the areas described by these definite integrals to determine whether your answers are reasonable.

Calculation 

Formula:

$\int x^{n}dx=\frac{x^{n+1}}{n+1}$

From the given integral:

$$\begin{gathered} {\int }_{-2}^2\left(x^2-1\right)dx \\ ={\left[\frac{x^3}{3}-x\right]}_{-2}^2 \\ =\left[\frac{2^3}{3}-2\right]-\left[\frac{{\left(-2\right)}^3}{3}-\left(-2\right)\right] \\ =\left[\frac{8}{3}-2\right]-\left[\frac{-8}{3}+2\right] \\ =\frac{4}{3} \end{gathered}$$

The sketch of the region is:

The area is finite, so our answer is reasonable.