Calculus Examples

$f (x) = x^{2} + x - 12$ ; $[0, 24]$

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 $[0, 24]$ .

$f (x)$ 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.

$A (x) = \frac{1}{24 - 0} (\int_{0}^{24} x^{2} + x - 12 d x)$

Step 5

Split the single integral into multiple integrals.

$A (x) = \frac{1}{24 - 0} (\int_{0}^{24} x^{2} d x + \int_{0}^{24} x d x + \int_{0}^{24} - 12 d x)$

Step 6

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

$A (x) = \frac{1}{24 - 0} (\frac{1}{3} x^{3}]_{0}^{24} + \int_{0}^{24} x d x + \int_{0}^{24} - 12 d x)$

Step 7

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

$A (x) = \frac{1}{24 - 0} (\frac{1}{3} x^{3}]_{0}^{24} + \frac{1}{2} x^{2}]_{0}^{24} + \int_{0}^{24} - 12 d x)$

Step 8

Apply the constant rule.

$A (x) = \frac{1}{24 - 0} (\frac{1}{3} x^{3}]_{0}^{24} + \frac{1}{2} x^{2}]_{0}^{24} + - 12 x]_{0}^{24})$

Step 9

Simplify the answer.

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Step 9.1

Simplify.

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Step 9.1.1

Combine $\frac{1}{3} x^{3}]_{0}^{24}$ and $\frac{1}{2} x^{2}]_{0}^{24}$ .

$A (x) = \frac{1}{24 - 0} (\frac{1}{3} x^{3} + \frac{1}{2} x^{2}]_{0}^{24} + - 12 x]_{0}^{24})$

Step 9.1.2

Combine $\frac{1}{3} x^{3} + \frac{1}{2} x^{2}]_{0}^{24}$ and $- 12 x]_{0}^{24}$ .

$A (x) = \frac{1}{24 - 0} (\frac{1}{3} x^{3} + \frac{1}{2} x^{2} - 12 x]_{0}^{24})$

Step 9.2

Substitute and simplify.

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Step 9.2.1

Evaluate $\frac{1}{3} x^{3} + \frac{1}{2} x^{2} - 12 x$ at $24$ and at $0$ .

$A (x) = \frac{1}{24 - 0} ((\frac{1}{3} \cdot 24^{3} + \frac{1}{2} \cdot 24^{2} - 12 \cdot 24) - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 12 \cdot 0))$

Step 9.2.2

Simplify.

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Step 9.2.2.1

Raise $24$ to the power of $3$ .

$A (x) = \frac{1}{24 - 0} (\frac{1}{3} \cdot 13824 + \frac{1}{2} \cdot 24^{2} - 12 \cdot 24 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 12 \cdot 0))$

Step 9.2.2.2

Combine $\frac{1}{3}$ and $13824$ .

$A (x) = \frac{1}{24 - 0} (\frac{13824}{3} + \frac{1}{2} \cdot 24^{2} - 12 \cdot 24 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 12 \cdot 0))$

Step 9.2.2.3

Cancel the common factor of $13824$ and $3$ .

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Step 9.2.2.3.1

Factor $3$ out of $13824$ .

$A (x) = \frac{1}{24 - 0} (\frac{3 \cdot 4608}{3} + \frac{1}{2} \cdot 24^{2} - 12 \cdot 24 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 12 \cdot 0))$

Step 9.2.2.3.2

Cancel the common factors.

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Step 9.2.2.3.2.1

Factor $3$ out of $3$ .

$A (x) = \frac{1}{24 - 0} (\frac{3 \cdot 4608}{3 (1)} + \frac{1}{2} \cdot 24^{2} - 12 \cdot 24 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 12 \cdot 0))$

Step 9.2.2.3.2.2

Cancel the common factor.

$A (x) = \frac{1}{24 - 0} (\frac{3 \cdot 4608}{3 \cdot 1} + \frac{1}{2} \cdot 24^{2} - 12 \cdot 24 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 12 \cdot 0))$

Step 9.2.2.3.2.3

Rewrite the expression.

$A (x) = \frac{1}{24 - 0} (\frac{4608}{1} + \frac{1}{2} \cdot 24^{2} - 12 \cdot 24 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 12 \cdot 0))$

Step 9.2.2.3.2.4

Divide $4608$ by $1$ .

$A (x) = \frac{1}{24 - 0} (4608 + \frac{1}{2} \cdot 24^{2} - 12 \cdot 24 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 12 \cdot 0))$

Step 9.2.2.4

Raise $24$ to the power of $2$ .

$A (x) = \frac{1}{24 - 0} (4608 + \frac{1}{2} \cdot 576 - 12 \cdot 24 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 12 \cdot 0))$

Step 9.2.2.5

Combine $\frac{1}{2}$ and $576$ .

$A (x) = \frac{1}{24 - 0} (4608 + \frac{576}{2} - 12 \cdot 24 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 12 \cdot 0))$

Step 9.2.2.6

Cancel the common factor of $576$ and $2$ .

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Step 9.2.2.6.1

Factor $2$ out of $576$ .

$A (x) = \frac{1}{24 - 0} (4608 + \frac{2 \cdot 288}{2} - 12 \cdot 24 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 12 \cdot 0))$

Step 9.2.2.6.2

Cancel the common factors.

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Step 9.2.2.6.2.1

Factor $2$ out of $2$ .

$A (x) = \frac{1}{24 - 0} (4608 + \frac{2 \cdot 288}{2 (1)} - 12 \cdot 24 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 12 \cdot 0))$

Step 9.2.2.6.2.2

Cancel the common factor.

$A (x) = \frac{1}{24 - 0} (4608 + \frac{2 \cdot 288}{2 \cdot 1} - 12 \cdot 24 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 12 \cdot 0))$

Step 9.2.2.6.2.3

Rewrite the expression.

$A (x) = \frac{1}{24 - 0} (4608 + \frac{288}{1} - 12 \cdot 24 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 12 \cdot 0))$

Step 9.2.2.6.2.4

Divide $288$ by $1$ .

$A (x) = \frac{1}{24 - 0} (4608 + 288 - 12 \cdot 24 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 12 \cdot 0))$

Step 9.2.2.7

Add $4608$ and $288$ .

$A (x) = \frac{1}{24 - 0} (4896 - 12 \cdot 24 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 12 \cdot 0))$

Step 9.2.2.8

Multiply $- 12$ by $24$ .

$A (x) = \frac{1}{24 - 0} (4896 - 288 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 12 \cdot 0))$

Step 9.2.2.9

Subtract $288$ from $4896$ .

$A (x) = \frac{1}{24 - 0} (4608 - (\frac{1}{3} \cdot 0^{3} + \frac{1}{2} \cdot 0^{2} - 12 \cdot 0))$

Step 9.2.2.10

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

$A (x) = \frac{1}{24 - 0} (4608 - (\frac{1}{3} \cdot 0 + \frac{1}{2} \cdot 0^{2} - 12 \cdot 0))$

Step 9.2.2.11

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

$A (x) = \frac{1}{24 - 0} (4608 - (0 + \frac{1}{2} \cdot 0^{2} - 12 \cdot 0))$

Step 9.2.2.12

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

$A (x) = \frac{1}{24 - 0} (4608 - (0 + \frac{1}{2} \cdot 0 - 12 \cdot 0))$

Step 9.2.2.13

Multiply $\frac{1}{2}$ by $0$ .

$A (x) = \frac{1}{24 - 0} (4608 - (0 + 0 - 12 \cdot 0))$

Step 9.2.2.14

Add $0$ and $0$ .

$A (x) = \frac{1}{24 - 0} (4608 - (0 - 12 \cdot 0))$

Step 9.2.2.15

Multiply $- 12$ by $0$ .

$A (x) = \frac{1}{24 - 0} (4608 - (0 + 0))$

Step 9.2.2.16

Add $0$ and $0$ .

$A (x) = \frac{1}{24 - 0} (4608 - 0)$

Step 9.2.2.17

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{24 - 0} (4608 + 0)$

Step 9.2.2.18

Add $4608$ and $0$ .

$A (x) = \frac{1}{24 - 0} (4608)$

Step 10

Simplify the denominator.

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Step 10.1

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{24 + 0} \cdot 4608$

Step 10.2

Add $24$ and $0$ .

$A (x) = \frac{1}{24} \cdot 4608$

Step 11

Cancel the common factor of $24$ .

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Step 11.1

Factor $24$ out of $4608$ .

$A (x) = \frac{1}{24} \cdot (24 (192))$

Step 11.2

Cancel the common factor.

$A (x) = \frac{1}{24} \cdot (24 \cdot 192)$

Step 11.3

Rewrite the expression.

$A (x) = 192$

Step 12