Calculus Examples

$f (x) = 5 x^{2} + 2 x - 3$ , $[- 3, 3]$

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 $[- 3, 3]$ .

$f (x)$ 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.

$A (x) = \frac{1}{3 + 3} (\int_{- 3}^{3} 5 x^{2} + 2 x - 3 d x)$

Step 5

Split the single integral into multiple integrals.

$A (x) = \frac{1}{3 + 3} (\int_{- 3}^{3} 5 x^{2} d x + \int_{- 3}^{3} 2 x d x + \int_{- 3}^{3} - 3 d x)$

Step 6

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$A (x) = \frac{1}{3 + 3} (5 \int_{- 3}^{3} x^{2} d x + \int_{- 3}^{3} 2 x d x + \int_{- 3}^{3} - 3 d x)$

Step 7

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

$A (x) = \frac{1}{3 + 3} (5 (\frac{1}{3} x^{3}]_{- 3}^{3}) + \int_{- 3}^{3} 2 x d x + \int_{- 3}^{3} - 3 d x)$

Step 8

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

$A (x) = \frac{1}{3 + 3} (5 (\frac{x^{3}}{3}]_{- 3}^{3}) + \int_{- 3}^{3} 2 x d x + \int_{- 3}^{3} - 3 d x)$

Step 9

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

$A (x) = \frac{1}{3 + 3} (5 (\frac{x^{3}}{3}]_{- 3}^{3}) + 2 \int_{- 3}^{3} x d x + \int_{- 3}^{3} - 3 d x)$

Step 10

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

$A (x) = \frac{1}{3 + 3} (5 (\frac{x^{3}}{3}]_{- 3}^{3}) + 2 (\frac{1}{2} x^{2}]_{- 3}^{3}) + \int_{- 3}^{3} - 3 d x)$

Step 11

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

$A (x) = \frac{1}{3 + 3} (5 (\frac{x^{3}}{3}]_{- 3}^{3}) + 2 (\frac{x^{2}}{2}]_{- 3}^{3}) + \int_{- 3}^{3} - 3 d x)$

Step 12

Apply the constant rule.

$A (x) = \frac{1}{3 + 3} (5 (\frac{x^{3}}{3}]_{- 3}^{3}) + 2 (\frac{x^{2}}{2}]_{- 3}^{3}) + - 3 x]_{- 3}^{3})$

Step 13

Substitute and simplify.

Tap for more steps...

Step 13.1

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

$A (x) = \frac{1}{3 + 3} (5 ((\frac{3^{3}}{3}) - \frac{{(- 3)}^{3}}{3}) + 2 (\frac{x^{2}}{2}]_{- 3}^{3}) + - 3 x]_{- 3}^{3})$

Step 13.2

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

$A (x) = \frac{1}{3 + 3} (5 (\frac{3^{3}}{3} - \frac{{(- 3)}^{3}}{3}) + 2 (\frac{3^{2}}{2} - \frac{{(- 3)}^{2}}{2}) + - 3 x]_{- 3}^{3})$

Step 13.3

Evaluate $- 3 x$ at $3$ and at $- 3$ .

$A (x) = \frac{1}{3 + 3} (5 (\frac{3^{3}}{3} - \frac{{(- 3)}^{3}}{3}) + 2 (\frac{3^{2}}{2} - \frac{{(- 3)}^{2}}{2}) - 3 \cdot 3 + 3 \cdot - 3)$

Step 13.4

Simplify.

Tap for more steps...

Step 13.4.1

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

$A (x) = \frac{1}{3 + 3} (5 (\frac{27}{3} - \frac{{(- 3)}^{3}}{3}) + 2 (\frac{3^{2}}{2} - \frac{{(- 3)}^{2}}{2}) - 3 \cdot 3 + 3 \cdot - 3)$

Step 13.4.2

Cancel the common factor of $27$ and $3$ .

Tap for more steps...

Step 13.4.2.1

Factor $3$ out of $27$ .

$A (x) = \frac{1}{3 + 3} (5 (\frac{3 \cdot 9}{3} - \frac{{(- 3)}^{3}}{3}) + 2 (\frac{3^{2}}{2} - \frac{{(- 3)}^{2}}{2}) - 3 \cdot 3 + 3 \cdot - 3)$

Step 13.4.2.2

Cancel the common factors.

Tap for more steps...

Step 13.4.2.2.1

Factor $3$ out of $3$ .

$A (x) = \frac{1}{3 + 3} (5 (\frac{3 \cdot 9}{3 (1)} - \frac{{(- 3)}^{3}}{3}) + 2 (\frac{3^{2}}{2} - \frac{{(- 3)}^{2}}{2}) - 3 \cdot 3 + 3 \cdot - 3)$

Step 13.4.2.2.2

Cancel the common factor.

$A (x) = \frac{1}{3 + 3} (5 (\frac{3 \cdot 9}{3 \cdot 1} - \frac{{(- 3)}^{3}}{3}) + 2 (\frac{3^{2}}{2} - \frac{{(- 3)}^{2}}{2}) - 3 \cdot 3 + 3 \cdot - 3)$

Step 13.4.2.2.3

Rewrite the expression.

$A (x) = \frac{1}{3 + 3} (5 (\frac{9}{1} - \frac{{(- 3)}^{3}}{3}) + 2 (\frac{3^{2}}{2} - \frac{{(- 3)}^{2}}{2}) - 3 \cdot 3 + 3 \cdot - 3)$

Step 13.4.2.2.4

Divide $9$ by $1$ .

$A (x) = \frac{1}{3 + 3} (5 (9 - \frac{{(- 3)}^{3}}{3}) + 2 (\frac{3^{2}}{2} - \frac{{(- 3)}^{2}}{2}) - 3 \cdot 3 + 3 \cdot - 3)$

Step 13.4.3

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

$A (x) = \frac{1}{3 + 3} (5 (9 - \frac{- 27}{3}) + 2 (\frac{3^{2}}{2} - \frac{{(- 3)}^{2}}{2}) - 3 \cdot 3 + 3 \cdot - 3)$

Step 13.4.4

Cancel the common factor of $- 27$ and $3$ .

Tap for more steps...

Step 13.4.4.1

Factor $3$ out of $- 27$ .

$A (x) = \frac{1}{3 + 3} (5 (9 - \frac{3 \cdot - 9}{3}) + 2 (\frac{3^{2}}{2} - \frac{{(- 3)}^{2}}{2}) - 3 \cdot 3 + 3 \cdot - 3)$

Step 13.4.4.2

Cancel the common factors.

Tap for more steps...

Step 13.4.4.2.1

Factor $3$ out of $3$ .

$A (x) = \frac{1}{3 + 3} (5 (9 - \frac{3 \cdot - 9}{3 (1)}) + 2 (\frac{3^{2}}{2} - \frac{{(- 3)}^{2}}{2}) - 3 \cdot 3 + 3 \cdot - 3)$

Step 13.4.4.2.2

Cancel the common factor.

$A (x) = \frac{1}{3 + 3} (5 (9 - \frac{3 \cdot - 9}{3 \cdot 1}) + 2 (\frac{3^{2}}{2} - \frac{{(- 3)}^{2}}{2}) - 3 \cdot 3 + 3 \cdot - 3)$

Step 13.4.4.2.3

Rewrite the expression.

$A (x) = \frac{1}{3 + 3} (5 (9 - \frac{- 9}{1}) + 2 (\frac{3^{2}}{2} - \frac{{(- 3)}^{2}}{2}) - 3 \cdot 3 + 3 \cdot - 3)$

Step 13.4.4.2.4

Divide $- 9$ by $1$ .

$A (x) = \frac{1}{3 + 3} (5 (9 + 9) + 2 (\frac{3^{2}}{2} - \frac{{(- 3)}^{2}}{2}) - 3 \cdot 3 + 3 \cdot - 3)$

Step 13.4.5

Multiply $- 1$ by $- 9$ .

$A (x) = \frac{1}{3 + 3} (5 (9 + 9) + 2 (\frac{3^{2}}{2} - \frac{{(- 3)}^{2}}{2}) - 3 \cdot 3 + 3 \cdot - 3)$

Step 13.4.6

Add $9$ and $9$ .

$A (x) = \frac{1}{3 + 3} (5 \cdot 18 + 2 (\frac{3^{2}}{2} - \frac{{(- 3)}^{2}}{2}) - 3 \cdot 3 + 3 \cdot - 3)$

Step 13.4.7

Multiply $5$ by $18$ .

$A (x) = \frac{1}{3 + 3} (90 + 2 (\frac{3^{2}}{2} - \frac{{(- 3)}^{2}}{2}) - 3 \cdot 3 + 3 \cdot - 3)$

Step 13.4.8

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

$A (x) = \frac{1}{3 + 3} (90 + 2 (\frac{9}{2} - \frac{{(- 3)}^{2}}{2}) - 3 \cdot 3 + 3 \cdot - 3)$

Step 13.4.9

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

$A (x) = \frac{1}{3 + 3} (90 + 2 (\frac{9}{2} - \frac{9}{2}) - 3 \cdot 3 + 3 \cdot - 3)$

Step 13.4.10

Combine the numerators over the common denominator.

$A (x) = \frac{1}{3 + 3} (90 + 2 (\frac{9 - 9}{2}) - 3 \cdot 3 + 3 \cdot - 3)$

Step 13.4.11

Subtract $9$ from $9$ .

$A (x) = \frac{1}{3 + 3} (90 + 2 (\frac{0}{2}) - 3 \cdot 3 + 3 \cdot - 3)$

Step 13.4.12

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

Tap for more steps...

Step 13.4.12.1

Factor $2$ out of $0$ .

$A (x) = \frac{1}{3 + 3} (90 + 2 (\frac{2 (0)}{2}) - 3 \cdot 3 + 3 \cdot - 3)$

Step 13.4.12.2

Cancel the common factors.

Tap for more steps...

Step 13.4.12.2.1

Factor $2$ out of $2$ .

$A (x) = \frac{1}{3 + 3} (90 + 2 (\frac{2 \cdot 0}{2 \cdot 1}) - 3 \cdot 3 + 3 \cdot - 3)$

Step 13.4.12.2.2

Cancel the common factor.

$A (x) = \frac{1}{3 + 3} (90 + 2 (\frac{2 \cdot 0}{2 \cdot 1}) - 3 \cdot 3 + 3 \cdot - 3)$

Step 13.4.12.2.3

Rewrite the expression.

$A (x) = \frac{1}{3 + 3} (90 + 2 (\frac{0}{1}) - 3 \cdot 3 + 3 \cdot - 3)$

Step 13.4.12.2.4

Divide $0$ by $1$ .

$A (x) = \frac{1}{3 + 3} (90 + 2 \cdot 0 - 3 \cdot 3 + 3 \cdot - 3)$

Step 13.4.13

Multiply $2$ by $0$ .

$A (x) = \frac{1}{3 + 3} (90 + 0 - 3 \cdot 3 + 3 \cdot - 3)$

Step 13.4.14

Add $90$ and $0$ .

$A (x) = \frac{1}{3 + 3} (90 - 3 \cdot 3 + 3 \cdot - 3)$

Step 13.4.15

Multiply $- 3$ by $3$ .

$A (x) = \frac{1}{3 + 3} (90 - 9 + 3 \cdot - 3)$

Step 13.4.16

Multiply $3$ by $- 3$ .

$A (x) = \frac{1}{3 + 3} (90 - 9 - 9)$

Step 13.4.17

Subtract $9$ from $- 9$ .

$A (x) = \frac{1}{3 + 3} (90 - 18)$

Step 13.4.18

Subtract $18$ from $90$ .

$A (x) = \frac{1}{3 + 3} (72)$

Step 14

Add $3$ and $3$ .

$A (x) = \frac{1}{6} \cdot 72$

Step 15

Cancel the common factor of $6$ .

Tap for more steps...

Step 15.1

Factor $6$ out of $72$ .

$A (x) = \frac{1}{6} \cdot (6 (12))$

Step 15.2

Cancel the common factor.

$A (x) = \frac{1}{6} \cdot (6 \cdot 12)$

Step 15.3

Rewrite the expression.

$A (x) = 12$

Step 16