Calculus Examples

$\int_{- 2}^{2} x^{2} d x$

Step 1

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$\frac{1}{3} x^{3}]_{- 2}^{2}$

Step 2

Substitute and simplify.

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Step 2.1

Evaluate $\frac{1}{3} x^{3}$ at $2$ and at $- 2$ .

$(\frac{1}{3} \cdot 2^{3}) - \frac{1}{3} {(- 2)}^{3}$

Step 2.2

Simplify.

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Step 2.2.1

Raise $2$ to the power of $3$ .

$\frac{1}{3} \cdot 8 - \frac{1}{3} {(- 2)}^{3}$

Step 2.2.2

Combine $\frac{1}{3}$ and $8$ .

$\frac{8}{3} - \frac{1}{3} {(- 2)}^{3}$

Step 2.2.3

Raise $- 2$ to the power of $3$ .

$\frac{8}{3} - \frac{1}{3} \cdot - 8$

Step 2.2.4

Multiply $- 8$ by $- 1$ .

$\frac{8}{3} + 8 (\frac{1}{3})$

Step 2.2.5

Combine $8$ and $\frac{1}{3}$ .

$\frac{8}{3} + \frac{8}{3}$

Step 2.2.6

Combine the numerators over the common denominator.

$\frac{8 + 8}{3}$

Step 2.2.7

Add $8$ and $8$ .

$\frac{16}{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{16}{3}$

Decimal Form:

$5.\overline{3}$

Mixed Number Form:

$5 \frac{1}{3}$

Step 4