Calculus Examples

$\int_{- 3}^{k} x^{2} d x = 0$

Step 1

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

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

Step 2

Substitute and simplify.

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Step 2.1

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

$(\frac{1}{3} k^{3}) - \frac{1}{3} {(- 3)}^{3}$

Step 2.2

Simplify.

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Step 2.2.1

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

$\frac{1}{3} k^{3} - \frac{1}{3} \cdot - 27$

Step 2.2.2

Multiply $- 27$ by $- 1$ .

$\frac{1}{3} k^{3} + 27 (\frac{1}{3})$

Step 2.2.3

Combine $27$ and $\frac{1}{3}$ .

$\frac{1}{3} k^{3} + \frac{27}{3}$

Step 2.2.4

Cancel the common factor of $27$ and $3$ .

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Step 2.2.4.1

Factor $3$ out of $27$ .

$\frac{1}{3} k^{3} + \frac{3 \cdot 9}{3}$

Step 2.2.4.2

Cancel the common factors.

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Step 2.2.4.2.1

Factor $3$ out of $3$ .

$\frac{1}{3} k^{3} + \frac{3 \cdot 9}{3 (1)}$

Step 2.2.4.2.2

Cancel the common factor.

$\frac{1}{3} k^{3} + \frac{3 \cdot 9}{3 \cdot 1}$

Step 2.2.4.2.3

Rewrite the expression.

$\frac{1}{3} k^{3} + \frac{9}{1}$

Step 2.2.4.2.4

Divide $9$ by $1$ .

$\frac{1}{3} k^{3} + 9$

Step 3

Combine $\frac{1}{3}$ and $k^{3}$ .

$\frac{k^{3}}{3} + 9$