Calculus Examples

$\int_{- 4}^{3} | \frac{1}{2} x | d x$

Step 1

Simplify.

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Step 1.1

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

$\int_{- 4}^{3} | \frac{x}{2} | d x$

Step 1.2

Remove non-negative terms from the absolute value.

$\int_{- 4}^{3} \frac{| x |}{2} d x$

Step 2

Since $\frac{1}{2}$ is constant with respect to $x$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int_{- 4}^{3} | x | d x$

Step 3

Split up the integral depending on where $x$ is positive and negative.

$\frac{1}{2} (\int_{- 4}^{0} - x d x + \int_{0}^{3} x d x)$

Step 4

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{1}{2} (- \int_{- 4}^{0} x d x + \int_{0}^{3} x d x)$

Step 5

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

$\frac{1}{2} (- (\frac{1}{2} x^{2}]_{- 4}^{0}) + \int_{0}^{3} x d x)$

Step 6

Combine $\frac{1}{2}$ and $x^{2}$ .

$\frac{1}{2} (- (\frac{x^{2}}{2}]_{- 4}^{0}) + \int_{0}^{3} x d x)$

Step 7

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

$\frac{1}{2} (- (\frac{x^{2}}{2}]_{- 4}^{0}) + \frac{1}{2} x^{2}]_{0}^{3})$

Step 8

Simplify the answer.

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Step 8.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$\frac{1}{2} (- (\frac{x^{2}}{2}]_{- 4}^{0}) + \frac{x^{2}}{2}]_{0}^{3})$

Step 8.2

Substitute and simplify.

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Step 8.2.1

Evaluate $\frac{x^{2}}{2}$ at $0$ and at $- 4$ .

$\frac{1}{2} (- ((\frac{0^{2}}{2}) - \frac{{(- 4)}^{2}}{2}) + \frac{x^{2}}{2}]_{0}^{3})$

Step 8.2.2

Evaluate $\frac{x^{2}}{2}$ at $3$ and at $0$ .

$\frac{1}{2} (- ((\frac{0^{2}}{2}) - \frac{{(- 4)}^{2}}{2}) + (\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3

Simplify.

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Step 8.2.3.1

Raising $0$ to any positive power yields $0$ .

$\frac{1}{2} (- (\frac{0}{2} - \frac{{(- 4)}^{2}}{2}) + (\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.2

Cancel the common factor of $0$ and $2$ .

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Step 8.2.3.2.1

Factor $2$ out of $0$ .

$\frac{1}{2} (- (\frac{2 (0)}{2} - \frac{{(- 4)}^{2}}{2}) + (\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.2.2

Cancel the common factors.

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Step 8.2.3.2.2.1

Factor $2$ out of $2$ .

$\frac{1}{2} (- (\frac{2 \cdot 0}{2 \cdot 1} - \frac{{(- 4)}^{2}}{2}) + (\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.2.2.2

Cancel the common factor.

$\frac{1}{2} (- (\frac{2 \cdot 0}{2 \cdot 1} - \frac{{(- 4)}^{2}}{2}) + (\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.2.2.3

Rewrite the expression.

$\frac{1}{2} (- (\frac{0}{1} - \frac{{(- 4)}^{2}}{2}) + (\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.2.2.4

Divide $0$ by $1$ .

$\frac{1}{2} (- (0 - \frac{{(- 4)}^{2}}{2}) + (\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.3

Raise $- 4$ to the power of $2$ .

$\frac{1}{2} (- (0 - \frac{16}{2}) + (\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.4

Cancel the common factor of $16$ and $2$ .

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Step 8.2.3.4.1

Factor $2$ out of $16$ .

$\frac{1}{2} (- (0 - \frac{2 \cdot 8}{2}) + (\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.4.2

Cancel the common factors.

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Step 8.2.3.4.2.1

Factor $2$ out of $2$ .

$\frac{1}{2} (- (0 - \frac{2 \cdot 8}{2 (1)}) + (\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.4.2.2

Cancel the common factor.

$\frac{1}{2} (- (0 - \frac{2 \cdot 8}{2 \cdot 1}) + (\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.4.2.3

Rewrite the expression.

$\frac{1}{2} (- (0 - \frac{8}{1}) + (\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.4.2.4

Divide $8$ by $1$ .

$\frac{1}{2} (- (0 - 1 \cdot 8) + (\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.5

Multiply $- 1$ by $8$ .

$\frac{1}{2} (- (0 - 8) + (\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.6

Subtract $8$ from $0$ .

$\frac{1}{2} (- - 8 + (\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.7

Multiply $- 1$ by $- 8$ .

$\frac{1}{2} (8 + (\frac{3^{2}}{2}) - \frac{0^{2}}{2})$

Step 8.2.3.8

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

$\frac{1}{2} (8 + \frac{9}{2} - \frac{0^{2}}{2})$

Step 8.2.3.9

Raising $0$ to any positive power yields $0$ .

$\frac{1}{2} (8 + \frac{9}{2} - \frac{0}{2})$

Step 8.2.3.10

Cancel the common factor of $0$ and $2$ .

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Step 8.2.3.10.1

Factor $2$ out of $0$ .

$\frac{1}{2} (8 + \frac{9}{2} - \frac{2 (0)}{2})$

Step 8.2.3.10.2

Cancel the common factors.

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Step 8.2.3.10.2.1

Factor $2$ out of $2$ .

$\frac{1}{2} (8 + \frac{9}{2} - \frac{2 \cdot 0}{2 \cdot 1})$

Step 8.2.3.10.2.2

Cancel the common factor.

$\frac{1}{2} (8 + \frac{9}{2} - \frac{2 \cdot 0}{2 \cdot 1})$

Step 8.2.3.10.2.3

Rewrite the expression.

$\frac{1}{2} (8 + \frac{9}{2} - \frac{0}{1})$

Step 8.2.3.10.2.4

Divide $0$ by $1$ .

$\frac{1}{2} (8 + \frac{9}{2} - 0)$

Step 8.2.3.11

Multiply $- 1$ by $0$ .

$\frac{1}{2} (8 + \frac{9}{2} + 0)$

Step 8.2.3.12

Add $\frac{9}{2}$ and $0$ .

$\frac{1}{2} (8 + \frac{9}{2})$

Step 8.2.3.13

To write $8$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} (8 \cdot \frac{2}{2} + \frac{9}{2})$

Step 8.2.3.14

Combine $8$ and $\frac{2}{2}$ .

$\frac{1}{2} (\frac{8 \cdot 2}{2} + \frac{9}{2})$

Step 8.2.3.15

Combine the numerators over the common denominator.

$\frac{1}{2} \cdot \frac{8 \cdot 2 + 9}{2}$

Step 8.2.3.16

Simplify the numerator.

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Step 8.2.3.16.1

Multiply $8$ by $2$ .

$\frac{1}{2} \cdot \frac{16 + 9}{2}$

Step 8.2.3.16.2

Add $16$ and $9$ .

$\frac{1}{2} \cdot \frac{25}{2}$

Step 8.2.3.17

Multiply $\frac{1}{2}$ by $\frac{25}{2}$ .

$\frac{25}{2 \cdot 2}$

Step 8.2.3.18

Multiply $2$ by $2$ .

$\frac{25}{4}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$\frac{25}{4}$

Decimal Form:

$6.25$

Mixed Number Form:

$6 \frac{1}{4}$

Step 10