Calculus Examples

$f (x) = x^{3} - 3 x^{2}$ , $[0, 4]$

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 $[0, 4]$ .

$f (x)$ 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.

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

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

Step 7

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

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

Step 8

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

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

Step 9

Simplify the answer.

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Step 9.1

Combine $\frac{1}{3}$ and $x^{3}$ .

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

Step 9.2

Substitute and simplify.

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Step 9.2.1

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

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

Step 9.2.2

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

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

Step 9.2.3

Simplify.

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Step 9.2.3.1

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

$A (x) = \frac{1}{4 - 0} (\frac{1}{4} \cdot 256 - \frac{1}{4} \cdot 0^{4} - 3 (\frac{4^{3}}{3} - \frac{0^{3}}{3}))$

Step 9.2.3.2

Combine $\frac{1}{4}$ and $256$ .

$A (x) = \frac{1}{4 - 0} (\frac{256}{4} - \frac{1}{4} \cdot 0^{4} - 3 (\frac{4^{3}}{3} - \frac{0^{3}}{3}))$

Step 9.2.3.3

Cancel the common factor of $256$ and $4$ .

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Step 9.2.3.3.1

Factor $4$ out of $256$ .

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

Step 9.2.3.3.2

Cancel the common factors.

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Step 9.2.3.3.2.1

Factor $4$ out of $4$ .

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

Step 9.2.3.3.2.2

Cancel the common factor.

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

Step 9.2.3.3.2.3

Rewrite the expression.

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

Step 9.2.3.3.2.4

Divide $64$ by $1$ .

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

Step 9.2.3.4

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

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

Step 9.2.3.5

Multiply $0$ by $- 1$ .

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

Step 9.2.3.6

Multiply $0$ by $\frac{1}{4}$ .

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

Step 9.2.3.7

Add $64$ and $0$ .

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

Step 9.2.3.8

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

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

Step 9.2.3.9

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

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

Step 9.2.3.10

Cancel the common factor of $0$ and $3$ .

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Step 9.2.3.10.1

Factor $3$ out of $0$ .

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

Step 9.2.3.10.2

Cancel the common factors.

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Step 9.2.3.10.2.1

Factor $3$ out of $3$ .

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

Step 9.2.3.10.2.2

Cancel the common factor.

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

Step 9.2.3.10.2.3

Rewrite the expression.

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

Step 9.2.3.10.2.4

Divide $0$ by $1$ .

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

Step 9.2.3.11

Multiply $- 1$ by $0$ .

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

Step 9.2.3.12

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

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

Step 9.2.3.13

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

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

Step 9.2.3.14

Multiply $- 3$ by $64$ .

$A (x) = \frac{1}{4 - 0} (64 + \frac{- 192}{3})$

Step 9.2.3.15

Cancel the common factor of $- 192$ and $3$ .

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Step 9.2.3.15.1

Factor $3$ out of $- 192$ .

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

Step 9.2.3.15.2

Cancel the common factors.

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Step 9.2.3.15.2.1

Factor $3$ out of $3$ .

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

Step 9.2.3.15.2.2

Cancel the common factor.

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

Step 9.2.3.15.2.3

Rewrite the expression.

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

Step 9.2.3.15.2.4

Divide $- 64$ by $1$ .

$A (x) = \frac{1}{4 - 0} (64 - 64)$

Step 9.2.3.16

Subtract $64$ from $64$ .

$A (x) = \frac{1}{4 - 0} (0)$

Step 10

Simplify the denominator.

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Step 10.1

Multiply $- 1$ by $0$ .

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

Step 10.2

Add $4$ and $0$ .

$A (x) = \frac{1}{4} \cdot 0$

Step 11

Multiply $\frac{1}{4}$ by $0$ .

$A (x) = 0$

Step 12