Calculus Examples

$\int_{1}^{3} x^{2} d x$

Step 1

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

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

Step 2

Substitute and simplify.

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Step 2.1

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

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

Step 2.2

Simplify.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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Step 2.2.3.1

Factor $3$ out of $27$ .

$\frac{3 \cdot 9}{3} - \frac{1}{3} \cdot 1^{3}$

Step 2.2.3.2

Cancel the common factors.

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Step 2.2.3.2.1

Factor $3$ out of $3$ .

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

Step 2.2.3.2.2

Cancel the common factor.

$\frac{3 \cdot 9}{3 \cdot 1} - \frac{1}{3} \cdot 1^{3}$

Step 2.2.3.2.3

Rewrite the expression.

$\frac{9}{1} - \frac{1}{3} \cdot 1^{3}$

Step 2.2.3.2.4

Divide $9$ by $1$ .

$9 - \frac{1}{3} \cdot 1^{3}$

Step 2.2.4

One to any power is one.

$9 - \frac{1}{3} \cdot 1$

Step 2.2.5

Multiply $- 1$ by $1$ .

$9 - \frac{1}{3}$

Step 2.2.6

To write $9$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$9 \cdot \frac{3}{3} - \frac{1}{3}$

Step 2.2.7

Combine $9$ and $\frac{3}{3}$ .

$\frac{9 \cdot 3}{3} - \frac{1}{3}$

Step 2.2.8

Combine the numerators over the common denominator.

$\frac{9 \cdot 3 - 1}{3}$

Step 2.2.9

Simplify the numerator.

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Step 2.2.9.1

Multiply $9$ by $3$ .

$\frac{27 - 1}{3}$

Step 2.2.9.2

Subtract $1$ from $27$ .

$\frac{26}{3}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{26}{3}$

Decimal Form:

$8.\overline{6}$

Mixed Number Form:

$8 \frac{2}{3}$

Step 4