Calculus Examples

$f (x) = x^{2} + 1$ , $- 1 < x < 2$

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 $[- 1, 2]$ .

$f (x)$ 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.

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

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

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

Step 7

Apply the constant rule.

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

Step 8

Simplify the answer.

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Step 8.1

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

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

Step 8.2

Substitute and simplify.

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Step 8.2.1

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

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

Step 8.2.2

Simplify.

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Step 8.2.2.1

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

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

Step 8.2.2.2

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

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

Step 8.2.2.3

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

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

Step 8.2.2.4

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

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

Step 8.2.2.5

Combine the numerators over the common denominator.

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

Step 8.2.2.6

Simplify the numerator.

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Step 8.2.2.6.1

Multiply $2$ by $3$ .

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

Step 8.2.2.6.2

Add $8$ and $6$ .

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

Step 8.2.2.7

Raise $- 1$ to the power of $3$ .

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

Step 8.2.2.8

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

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

Step 8.2.2.9

Move the negative in front of the fraction.

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

Step 8.2.2.10

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 8.2.2.11

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

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

Step 8.2.2.12

Combine the numerators over the common denominator.

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

Step 8.2.2.13

Simplify the numerator.

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Step 8.2.2.13.1

Multiply $- 1$ by $3$ .

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

Step 8.2.2.13.2

Subtract $3$ from $- 1$ .

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

Step 8.2.2.14

Move the negative in front of the fraction.

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

Step 8.2.2.15

Multiply $- 1$ by $- 1$ .

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

Step 8.2.2.16

Multiply $\frac{4}{3}$ by $1$ .

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

Step 8.2.2.17

Combine the numerators over the common denominator.

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

Step 8.2.2.18

Add $14$ and $4$ .

$A (x) = \frac{1}{2 + 1} (\frac{18}{3})$

Step 8.2.2.19

Cancel the common factor of $18$ and $3$ .

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Step 8.2.2.19.1

Factor $3$ out of $18$ .

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

Step 8.2.2.19.2

Cancel the common factors.

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Step 8.2.2.19.2.1

Factor $3$ out of $3$ .

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

Step 8.2.2.19.2.2

Cancel the common factor.

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

Step 8.2.2.19.2.3

Rewrite the expression.

$A (x) = \frac{1}{2 + 1} (\frac{6}{1})$

Step 8.2.2.19.2.4

Divide $6$ by $1$ .

$A (x) = \frac{1}{2 + 1} (6)$

Step 9

Add $2$ and $1$ .

$A (x) = \frac{1}{3} \cdot 6$

Step 10

Cancel the common factor of $3$ .

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Step 10.1

Factor $3$ out of $6$ .

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

Step 10.2

Cancel the common factor.

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

Step 10.3

Rewrite the expression.

$A (x) = 2$

Step 11