Calculus Examples

$f (x) = \frac{x^{2}}{2} + 2 x + 2$ , $[- 4, 0]$

Step 1

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 2

$f (x)$ is continuous on $[- 4, 0]$ .

$f (x)$ is continuous

Step 3

The average value of function $f$ over the interval $[a, b]$ is defined as $A (x) = \frac{1}{b - a} \int_{a}^{b} f (x) d x$ .

$A (x) = \frac{1}{b - a} \int_{a}^{b} f (x) d x$

Step 4

Substitute the actual values into the formula for the average value of a function.

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

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

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

Step 7

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

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

Step 8

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

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

Step 9

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

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

Step 10

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

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

Step 11

Apply the constant rule.

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

Step 12

Substitute and simplify.

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Step 12.1

Evaluate $\frac{1}{3} x^{3}$ at $0$ and at $- 4$ .

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

Step 12.2

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

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

Step 12.3

Evaluate $2 x$ at $0$ and at $- 4$ .

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

Step 12.4

Simplify.

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Step 12.4.1

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

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

Step 12.4.2

Multiply $\frac{1}{3}$ by $0$ .

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

Step 12.4.3

Raise $- 4$ to the power of $3$ .

$A (x) = \frac{1}{0 + 4} (\frac{1}{2} \cdot (0 - \frac{1}{3} \cdot - 64) + 2 (\frac{0^{2}}{2} - \frac{{(- 4)}^{2}}{2}) + 2 \cdot 0 - 2 \cdot - 4)$

Step 12.4.4

Multiply $- 64$ by $- 1$ .

$A (x) = \frac{1}{0 + 4} (\frac{1}{2} \cdot (0 + 64 (\frac{1}{3})) + 2 (\frac{0^{2}}{2} - \frac{{(- 4)}^{2}}{2}) + 2 \cdot 0 - 2 \cdot - 4)$

Step 12.4.5

Combine $64$ and $\frac{1}{3}$ .

$A (x) = \frac{1}{0 + 4} (\frac{1}{2} \cdot (0 + \frac{64}{3}) + 2 (\frac{0^{2}}{2} - \frac{{(- 4)}^{2}}{2}) + 2 \cdot 0 - 2 \cdot - 4)$

Step 12.4.6

Add $0$ and $\frac{64}{3}$ .

$A (x) = \frac{1}{0 + 4} (\frac{1}{2} \cdot \frac{64}{3} + 2 (\frac{0^{2}}{2} - \frac{{(- 4)}^{2}}{2}) + 2 \cdot 0 - 2 \cdot - 4)$

Step 12.4.7

Multiply $\frac{1}{2}$ by $\frac{64}{3}$ .

$A (x) = \frac{1}{0 + 4} (\frac{64}{2 \cdot 3} + 2 (\frac{0^{2}}{2} - \frac{{(- 4)}^{2}}{2}) + 2 \cdot 0 - 2 \cdot - 4)$

Step 12.4.8

Multiply $2$ by $3$ .

$A (x) = \frac{1}{0 + 4} (\frac{64}{6} + 2 (\frac{0^{2}}{2} - \frac{{(- 4)}^{2}}{2}) + 2 \cdot 0 - 2 \cdot - 4)$

Step 12.4.9

Cancel the common factor of $64$ and $6$ .

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Step 12.4.9.1

Factor $2$ out of $64$ .

$A (x) = \frac{1}{0 + 4} (\frac{2 (32)}{6} + 2 (\frac{0^{2}}{2} - \frac{{(- 4)}^{2}}{2}) + 2 \cdot 0 - 2 \cdot - 4)$

Step 12.4.9.2

Cancel the common factors.

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Step 12.4.9.2.1

Factor $2$ out of $6$ .

$A (x) = \frac{1}{0 + 4} (\frac{2 \cdot 32}{2 \cdot 3} + 2 (\frac{0^{2}}{2} - \frac{{(- 4)}^{2}}{2}) + 2 \cdot 0 - 2 \cdot - 4)$

Step 12.4.9.2.2

Cancel the common factor.

$A (x) = \frac{1}{0 + 4} (\frac{2 \cdot 32}{2 \cdot 3} + 2 (\frac{0^{2}}{2} - \frac{{(- 4)}^{2}}{2}) + 2 \cdot 0 - 2 \cdot - 4)$

Step 12.4.9.2.3

Rewrite the expression.

$A (x) = \frac{1}{0 + 4} (\frac{32}{3} + 2 (\frac{0^{2}}{2} - \frac{{(- 4)}^{2}}{2}) + 2 \cdot 0 - 2 \cdot - 4)$

Step 12.4.10

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

$A (x) = \frac{1}{0 + 4} (\frac{32}{3} + 2 (\frac{0}{2} - \frac{{(- 4)}^{2}}{2}) + 2 \cdot 0 - 2 \cdot - 4)$

Step 12.4.11

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

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Step 12.4.11.1

Factor $2$ out of $0$ .

$A (x) = \frac{1}{0 + 4} (\frac{32}{3} + 2 (\frac{2 (0)}{2} - \frac{{(- 4)}^{2}}{2}) + 2 \cdot 0 - 2 \cdot - 4)$

Step 12.4.11.2

Cancel the common factors.

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Step 12.4.11.2.1

Factor $2$ out of $2$ .

$A (x) = \frac{1}{0 + 4} (\frac{32}{3} + 2 (\frac{2 \cdot 0}{2 \cdot 1} - \frac{{(- 4)}^{2}}{2}) + 2 \cdot 0 - 2 \cdot - 4)$

Step 12.4.11.2.2

Cancel the common factor.

$A (x) = \frac{1}{0 + 4} (\frac{32}{3} + 2 (\frac{2 \cdot 0}{2 \cdot 1} - \frac{{(- 4)}^{2}}{2}) + 2 \cdot 0 - 2 \cdot - 4)$

Step 12.4.11.2.3

Rewrite the expression.

$A (x) = \frac{1}{0 + 4} (\frac{32}{3} + 2 (\frac{0}{1} - \frac{{(- 4)}^{2}}{2}) + 2 \cdot 0 - 2 \cdot - 4)$

Step 12.4.11.2.4

Divide $0$ by $1$ .

$A (x) = \frac{1}{0 + 4} (\frac{32}{3} + 2 (0 - \frac{{(- 4)}^{2}}{2}) + 2 \cdot 0 - 2 \cdot - 4)$

Step 12.4.12

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

$A (x) = \frac{1}{0 + 4} (\frac{32}{3} + 2 (0 - \frac{16}{2}) + 2 \cdot 0 - 2 \cdot - 4)$

Step 12.4.13

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

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Step 12.4.13.1

Factor $2$ out of $16$ .

$A (x) = \frac{1}{0 + 4} (\frac{32}{3} + 2 (0 - \frac{2 \cdot 8}{2}) + 2 \cdot 0 - 2 \cdot - 4)$

Step 12.4.13.2

Cancel the common factors.

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Step 12.4.13.2.1

Factor $2$ out of $2$ .

$A (x) = \frac{1}{0 + 4} (\frac{32}{3} + 2 (0 - \frac{2 \cdot 8}{2 (1)}) + 2 \cdot 0 - 2 \cdot - 4)$

Step 12.4.13.2.2

Cancel the common factor.

$A (x) = \frac{1}{0 + 4} (\frac{32}{3} + 2 (0 - \frac{2 \cdot 8}{2 \cdot 1}) + 2 \cdot 0 - 2 \cdot - 4)$

Step 12.4.13.2.3

Rewrite the expression.

$A (x) = \frac{1}{0 + 4} (\frac{32}{3} + 2 (0 - \frac{8}{1}) + 2 \cdot 0 - 2 \cdot - 4)$

Step 12.4.13.2.4

Divide $8$ by $1$ .

$A (x) = \frac{1}{0 + 4} (\frac{32}{3} + 2 (0 - 1 \cdot 8) + 2 \cdot 0 - 2 \cdot - 4)$

Step 12.4.14

Multiply $- 1$ by $8$ .

$A (x) = \frac{1}{0 + 4} (\frac{32}{3} + 2 (0 - 8) + 2 \cdot 0 - 2 \cdot - 4)$

Step 12.4.15

Subtract $8$ from $0$ .

$A (x) = \frac{1}{0 + 4} (\frac{32}{3} + 2 \cdot - 8 + 2 \cdot 0 - 2 \cdot - 4)$

Step 12.4.16

Multiply $2$ by $- 8$ .

$A (x) = \frac{1}{0 + 4} (\frac{32}{3} - 16 + 2 \cdot 0 - 2 \cdot - 4)$

Step 12.4.17

To write $- 16$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$A (x) = \frac{1}{0 + 4} (\frac{32}{3} - 16 \cdot \frac{3}{3} + 2 \cdot 0 - 2 \cdot - 4)$

Step 12.4.18

Combine $- 16$ and $\frac{3}{3}$ .

$A (x) = \frac{1}{0 + 4} (\frac{32}{3} + \frac{- 16 \cdot 3}{3} + 2 \cdot 0 - 2 \cdot - 4)$

Step 12.4.19

Combine the numerators over the common denominator.

$A (x) = \frac{1}{0 + 4} (\frac{32 - 16 \cdot 3}{3} + 2 \cdot 0 - 2 \cdot - 4)$

Step 12.4.20

Simplify the numerator.

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Step 12.4.20.1

Multiply $- 16$ by $3$ .

$A (x) = \frac{1}{0 + 4} (\frac{32 - 48}{3} + 2 \cdot 0 - 2 \cdot - 4)$

Step 12.4.20.2

Subtract $48$ from $32$ .

$A (x) = \frac{1}{0 + 4} (\frac{- 16}{3} + 2 \cdot 0 - 2 \cdot - 4)$

Step 12.4.21

Move the negative in front of the fraction.

$A (x) = \frac{1}{0 + 4} (- \frac{16}{3} + 2 \cdot 0 - 2 \cdot - 4)$

Step 12.4.22

Multiply $2$ by $0$ .

$A (x) = \frac{1}{0 + 4} (- \frac{16}{3} + 0 - 2 \cdot - 4)$

Step 12.4.23

Multiply $- 2$ by $- 4$ .

$A (x) = \frac{1}{0 + 4} (- \frac{16}{3} + 0 + 8)$

Step 12.4.24

Add $0$ and $8$ .

$A (x) = \frac{1}{0 + 4} (- \frac{16}{3} + 8)$

Step 12.4.25

To write $8$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$A (x) = \frac{1}{0 + 4} (- \frac{16}{3} + 8 \cdot \frac{3}{3})$

Step 12.4.26

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

$A (x) = \frac{1}{0 + 4} (- \frac{16}{3} + \frac{8 \cdot 3}{3})$

Step 12.4.27

Combine the numerators over the common denominator.

$A (x) = \frac{1}{0 + 4} (\frac{- 16 + 8 \cdot 3}{3})$

Step 12.4.28

Simplify the numerator.

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Step 12.4.28.1

Multiply $8$ by $3$ .

$A (x) = \frac{1}{0 + 4} (\frac{- 16 + 24}{3})$

Step 12.4.28.2

Add $- 16$ and $24$ .

$A (x) = \frac{1}{0 + 4} (\frac{8}{3})$

Step 13

Add $0$ and $4$ .

$A (x) = \frac{1}{4} \cdot \frac{8}{3}$

Step 14

Cancel the common factor of $4$ .

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Step 14.1

Factor $4$ out of $8$ .

$A (x) = \frac{1}{4} \cdot \frac{4 (2)}{3}$

Step 14.2

Cancel the common factor.

$A (x) = \frac{1}{4} \cdot \frac{4 \cdot 2}{3}$

Step 14.3

Rewrite the expression.

$A (x) = \frac{2}{3}$

Step 15