Calculus Examples

$f (x) = x^{2} - 2 x + 1$ , $[0, 2]$

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 $[0, 2]$ .

$f (x)$ 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.

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

Apply the constant rule.

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

Step 11

Simplify the answer.

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Step 11.1

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

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

Step 11.2

Substitute and simplify.

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Step 11.2.1

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

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

Step 11.2.2

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

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

Step 11.2.3

Simplify.

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Step 11.2.3.1

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

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

Step 11.2.3.2

Combine $\frac{1}{3}$ and $8$ .

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

Step 11.2.3.3

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 11.2.3.4

Combine $2$ and $\frac{3}{3}$ .

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

Step 11.2.3.5

Combine the numerators over the common denominator.

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

Step 11.2.3.6

Simplify the numerator.

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Step 11.2.3.6.1

Multiply $2$ by $3$ .

$A (x) = \frac{1}{2 - 0} (\frac{8 + 6}{3} - (\frac{1}{3} \cdot 0^{3} + 0) - 2 ((\frac{2^{2}}{2}) - \frac{0^{2}}{2}))$

Step 11.2.3.6.2

Add $8$ and $6$ .

$A (x) = \frac{1}{2 - 0} (\frac{14}{3} - (\frac{1}{3} \cdot 0^{3} + 0) - 2 ((\frac{2^{2}}{2}) - \frac{0^{2}}{2}))$

Step 11.2.3.7

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

$A (x) = \frac{1}{2 - 0} (\frac{14}{3} - (\frac{1}{3} \cdot 0 + 0) - 2 ((\frac{2^{2}}{2}) - \frac{0^{2}}{2}))$

Step 11.2.3.8

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

$A (x) = \frac{1}{2 - 0} (\frac{14}{3} - (0 + 0) - 2 ((\frac{2^{2}}{2}) - \frac{0^{2}}{2}))$

Step 11.2.3.9

Add $0$ and $0$ .

$A (x) = \frac{1}{2 - 0} (\frac{14}{3} - 0 - 2 ((\frac{2^{2}}{2}) - \frac{0^{2}}{2}))$

Step 11.2.3.10

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{2 - 0} (\frac{14}{3} + 0 - 2 ((\frac{2^{2}}{2}) - \frac{0^{2}}{2}))$

Step 11.2.3.11

Add $\frac{14}{3}$ and $0$ .

$A (x) = \frac{1}{2 - 0} (\frac{14}{3} - 2 ((\frac{2^{2}}{2}) - \frac{0^{2}}{2}))$

Step 11.2.3.12

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

$A (x) = \frac{1}{2 - 0} (\frac{14}{3} - 2 (\frac{4}{2} - \frac{0^{2}}{2}))$

Step 11.2.3.13

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

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Step 11.2.3.13.1

Factor $2$ out of $4$ .

$A (x) = \frac{1}{2 - 0} (\frac{14}{3} - 2 (\frac{2 \cdot 2}{2} - \frac{0^{2}}{2}))$

Step 11.2.3.13.2

Cancel the common factors.

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Step 11.2.3.13.2.1

Factor $2$ out of $2$ .

$A (x) = \frac{1}{2 - 0} (\frac{14}{3} - 2 (\frac{2 \cdot 2}{2 (1)} - \frac{0^{2}}{2}))$

Step 11.2.3.13.2.2

Cancel the common factor.

$A (x) = \frac{1}{2 - 0} (\frac{14}{3} - 2 (\frac{2 \cdot 2}{2 \cdot 1} - \frac{0^{2}}{2}))$

Step 11.2.3.13.2.3

Rewrite the expression.

$A (x) = \frac{1}{2 - 0} (\frac{14}{3} - 2 (\frac{2}{1} - \frac{0^{2}}{2}))$

Step 11.2.3.13.2.4

Divide $2$ by $1$ .

$A (x) = \frac{1}{2 - 0} (\frac{14}{3} - 2 (2 - \frac{0^{2}}{2}))$

Step 11.2.3.14

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

$A (x) = \frac{1}{2 - 0} (\frac{14}{3} - 2 (2 - \frac{0}{2}))$

Step 11.2.3.15

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

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Step 11.2.3.15.1

Factor $2$ out of $0$ .

$A (x) = \frac{1}{2 - 0} (\frac{14}{3} - 2 (2 - \frac{2 (0)}{2}))$

Step 11.2.3.15.2

Cancel the common factors.

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Step 11.2.3.15.2.1

Factor $2$ out of $2$ .

$A (x) = \frac{1}{2 - 0} (\frac{14}{3} - 2 (2 - \frac{2 \cdot 0}{2 \cdot 1}))$

Step 11.2.3.15.2.2

Cancel the common factor.

$A (x) = \frac{1}{2 - 0} (\frac{14}{3} - 2 (2 - \frac{2 \cdot 0}{2 \cdot 1}))$

Step 11.2.3.15.2.3

Rewrite the expression.

$A (x) = \frac{1}{2 - 0} (\frac{14}{3} - 2 (2 - \frac{0}{1}))$

Step 11.2.3.15.2.4

Divide $0$ by $1$ .

$A (x) = \frac{1}{2 - 0} (\frac{14}{3} - 2 (2 - 0))$

Step 11.2.3.16

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{2 - 0} (\frac{14}{3} - 2 (2 + 0))$

Step 11.2.3.17

Add $2$ and $0$ .

$A (x) = \frac{1}{2 - 0} (\frac{14}{3} - 2 \cdot 2)$

Step 11.2.3.18

Multiply $- 2$ by $2$ .

$A (x) = \frac{1}{2 - 0} (\frac{14}{3} - 4)$

Step 11.2.3.19

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$A (x) = \frac{1}{2 - 0} (\frac{14}{3} - 4 \cdot \frac{3}{3})$

Step 11.2.3.20

Combine $- 4$ and $\frac{3}{3}$ .

$A (x) = \frac{1}{2 - 0} (\frac{14}{3} + \frac{- 4 \cdot 3}{3})$

Step 11.2.3.21

Combine the numerators over the common denominator.

$A (x) = \frac{1}{2 - 0} (\frac{14 - 4 \cdot 3}{3})$

Step 11.2.3.22

Simplify the numerator.

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Step 11.2.3.22.1

Multiply $- 4$ by $3$ .

$A (x) = \frac{1}{2 - 0} (\frac{14 - 12}{3})$

Step 11.2.3.22.2

Subtract $12$ from $14$ .

$A (x) = \frac{1}{2 - 0} (\frac{2}{3})$

Step 12

Simplify the denominator.

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Step 12.1

Multiply $- 1$ by $0$ .

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

Step 12.2

Add $2$ and $0$ .

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

Step 13

Cancel the common factor of $2$ .

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Step 13.1

Cancel the common factor.

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

Step 13.2

Rewrite the expression.

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

Step 14