Calculus Examples

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

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 $[0, 3]$ .

$f (x)$ 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.

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

Step 5

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

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

Step 6

Substitute and simplify.

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Step 6.1

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

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

Step 6.2

Simplify.

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Step 6.2.1

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

$A (x) = \frac{1}{3 - 0} (\frac{1}{3} \cdot 27 - \frac{1}{3} \cdot 0^{3})$

Step 6.2.2

Combine $\frac{1}{3}$ and $27$ .

$A (x) = \frac{1}{3 - 0} (\frac{27}{3} - \frac{1}{3} \cdot 0^{3})$

Step 6.2.3

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

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Step 6.2.3.1

Factor $3$ out of $27$ .

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

Step 6.2.3.2

Cancel the common factors.

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Step 6.2.3.2.1

Factor $3$ out of $3$ .

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

Step 6.2.3.2.2

Cancel the common factor.

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

Step 6.2.3.2.3

Rewrite the expression.

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

Step 6.2.3.2.4

Divide $9$ by $1$ .

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

Step 6.2.4

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

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

Step 6.2.5

Multiply $0$ by $- 1$ .

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

Step 6.2.6

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

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

Step 6.2.7

Add $9$ and $0$ .

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

Step 7

Simplify the denominator.

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Step 7.1

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{3 + 0} \cdot 9$

Step 7.2

Add $3$ and $0$ .

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

Step 8

Cancel the common factor of $3$ .

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Step 8.1

Factor $3$ out of $9$ .

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

Step 8.2

Cancel the common factor.

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

Step 8.3

Rewrite the expression.

$A (x) = 3$

Step 9