Calculus Examples

$\int_{π}^{2 π} θ d θ$

Step 1

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

$\frac{1}{2} θ^{2}]_{π}^{2 π}$

Step 2

Substitute and simplify.

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Step 2.1

Evaluate $\frac{1}{2} θ^{2}$ at $2 π$ and at $π$ .

$(\frac{1}{2} {(2 π)}^{2}) - \frac{1}{2} π^{2}$

Step 2.2

Simplify.

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Step 2.2.1

Factor $2$ out of $2 π$ .

$\frac{1}{2} {(2 (π))}^{2} - \frac{1}{2} π^{2}$

Step 2.2.2

Apply the product rule to $2 (π)$ .

$\frac{1}{2} (2^{2} π^{2}) - \frac{1}{2} π^{2}$

Step 2.2.3

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

$\frac{1}{2} (4 π^{2}) - \frac{1}{2} π^{2}$

Step 2.2.4

Combine $4$ and $\frac{1}{2}$ .

$\frac{4}{2} π^{2} - \frac{1}{2} π^{2}$

Step 2.2.5

Combine $\frac{4}{2}$ and $π^{2}$ .

$\frac{4 π^{2}}{2} - \frac{1}{2} π^{2}$

Step 2.2.6

Cancel the common factor of $4$ and $2$ .

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Step 2.2.6.1

Factor $2$ out of $4 π^{2}$ .

$\frac{2 (2 π^{2})}{2} - \frac{1}{2} π^{2}$

Step 2.2.6.2

Cancel the common factors.

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Step 2.2.6.2.1

Factor $2$ out of $2$ .

$\frac{2 (2 π^{2})}{2 (1)} - \frac{1}{2} π^{2}$

Step 2.2.6.2.2

Cancel the common factor.

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

Step 2.2.6.2.3

Rewrite the expression.

$\frac{2 π^{2}}{1} - \frac{1}{2} π^{2}$

Step 2.2.6.2.4

Divide $2 π^{2}$ by $1$ .

$2 π^{2} - \frac{1}{2} π^{2}$

Step 2.2.7

Combine $π^{2}$ and $\frac{1}{2}$ .

$2 π^{2} - \frac{π^{2}}{2}$

Step 2.2.8

To write $2 π^{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 π^{2} \cdot \frac{2}{2} - \frac{π^{2}}{2}$

Step 2.2.9

Combine $2 π^{2}$ and $\frac{2}{2}$ .

$\frac{2 π^{2} \cdot 2}{2} - \frac{π^{2}}{2}$

Step 2.2.10

Combine the numerators over the common denominator.

$\frac{2 π^{2} \cdot 2 - π^{2}}{2}$

Step 2.2.11

Multiply $2$ by $2$ .

$\frac{4 π^{2} - π^{2}}{2}$

Step 2.2.12

Subtract $π^{2}$ from $4 π^{2}$ .

$\frac{3 π^{2}}{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$\frac{3 π^{2}}{2}$

Decimal Form:

$14.80440660 \dots$

Step 4