Calculus Examples

$f (x) = x^{2} - 7$ , $[0, 9]$

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 $[0, 9]$ .

$f (x)$ 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.

$A (x) = \frac{1}{9 - 0} (\int_{0}^{9} x^{2} - 7 d x)$

Step 5

Split the single integral into multiple integrals.

$A (x) = \frac{1}{9 - 0} (\int_{0}^{9} x^{2} d x + \int_{0}^{9} - 7 d x)$

Step 6

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

$A (x) = \frac{1}{9 - 0} (\frac{1}{3} x^{3}]_{0}^{9} + \int_{0}^{9} - 7 d x)$

Step 7

Apply the constant rule.

$A (x) = \frac{1}{9 - 0} (\frac{1}{3} x^{3}]_{0}^{9} + - 7 x]_{0}^{9})$

Step 8

Simplify the answer.

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Step 8.1

Combine $\frac{1}{3} x^{3}]_{0}^{9}$ and $- 7 x]_{0}^{9}$ .

$A (x) = \frac{1}{9 - 0} (\frac{1}{3} x^{3} - 7 x]_{0}^{9})$

Step 8.2

Substitute and simplify.

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Step 8.2.1

Evaluate $\frac{1}{3} x^{3} - 7 x$ at $9$ and at $0$ .

$A (x) = \frac{1}{9 - 0} ((\frac{1}{3} \cdot 9^{3} - 7 \cdot 9) - (\frac{1}{3} \cdot 0^{3} - 7 \cdot 0))$

Step 8.2.2

Simplify.

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Step 8.2.2.1

Raise $9$ to the power of $3$ .

$A (x) = \frac{1}{9 - 0} (\frac{1}{3} \cdot 729 - 7 \cdot 9 - (\frac{1}{3} \cdot 0^{3} - 7 \cdot 0))$

Step 8.2.2.2

Combine $\frac{1}{3}$ and $729$ .

$A (x) = \frac{1}{9 - 0} (\frac{729}{3} - 7 \cdot 9 - (\frac{1}{3} \cdot 0^{3} - 7 \cdot 0))$

Step 8.2.2.3

Cancel the common factor of $729$ and $3$ .

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Step 8.2.2.3.1

Factor $3$ out of $729$ .

$A (x) = \frac{1}{9 - 0} (\frac{3 \cdot 243}{3} - 7 \cdot 9 - (\frac{1}{3} \cdot 0^{3} - 7 \cdot 0))$

Step 8.2.2.3.2

Cancel the common factors.

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Step 8.2.2.3.2.1

Factor $3$ out of $3$ .

$A (x) = \frac{1}{9 - 0} (\frac{3 \cdot 243}{3 (1)} - 7 \cdot 9 - (\frac{1}{3} \cdot 0^{3} - 7 \cdot 0))$

Step 8.2.2.3.2.2

Cancel the common factor.

$A (x) = \frac{1}{9 - 0} (\frac{3 \cdot 243}{3 \cdot 1} - 7 \cdot 9 - (\frac{1}{3} \cdot 0^{3} - 7 \cdot 0))$

Step 8.2.2.3.2.3

Rewrite the expression.

$A (x) = \frac{1}{9 - 0} (\frac{243}{1} - 7 \cdot 9 - (\frac{1}{3} \cdot 0^{3} - 7 \cdot 0))$

Step 8.2.2.3.2.4

Divide $243$ by $1$ .

$A (x) = \frac{1}{9 - 0} (243 - 7 \cdot 9 - (\frac{1}{3} \cdot 0^{3} - 7 \cdot 0))$

Step 8.2.2.4

Multiply $- 7$ by $9$ .

$A (x) = \frac{1}{9 - 0} (243 - 63 - (\frac{1}{3} \cdot 0^{3} - 7 \cdot 0))$

Step 8.2.2.5

Subtract $63$ from $243$ .

$A (x) = \frac{1}{9 - 0} (180 - (\frac{1}{3} \cdot 0^{3} - 7 \cdot 0))$

Step 8.2.2.6

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

$A (x) = \frac{1}{9 - 0} (180 - (\frac{1}{3} \cdot 0 - 7 \cdot 0))$

Step 8.2.2.7

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

$A (x) = \frac{1}{9 - 0} (180 - (0 - 7 \cdot 0))$

Step 8.2.2.8

Multiply $- 7$ by $0$ .

$A (x) = \frac{1}{9 - 0} (180 - (0 + 0))$

Step 8.2.2.9

Add $0$ and $0$ .

$A (x) = \frac{1}{9 - 0} (180 - 0)$

Step 8.2.2.10

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{9 - 0} (180 + 0)$

Step 8.2.2.11

Add $180$ and $0$ .

$A (x) = \frac{1}{9 - 0} (180)$

Step 9

Simplify the denominator.

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Step 9.1

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{9 + 0} \cdot 180$

Step 9.2

Add $9$ and $0$ .

$A (x) = \frac{1}{9} \cdot 180$

Step 10

Cancel the common factor of $9$ .

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Step 10.1

Factor $9$ out of $180$ .

$A (x) = \frac{1}{9} \cdot (9 (20))$

Step 10.2

Cancel the common factor.

$A (x) = \frac{1}{9} \cdot (9 \cdot 20)$

Step 10.3

Rewrite the expression.

$A (x) = 20$

Step 11