Calculus Examples

$f (x) = 6 x$ , $[1, 5]$

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 $[1, 5]$ .

$f (x)$ 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.

$A (x) = \frac{1}{5 - 1} (\int_{1}^{5} 6 x d x)$

Step 5

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$A (x) = \frac{1}{5 - 1} (6 \int_{1}^{5} x d x)$

Step 6

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

$A (x) = \frac{1}{5 - 1} (6 (\frac{1}{2} x^{2}]_{1}^{5}))$

Step 7

Simplify the answer.

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Step 7.1

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

$A (x) = \frac{1}{5 - 1} (6 (\frac{x^{2}}{2}]_{1}^{5}))$

Step 7.2

Substitute and simplify.

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Step 7.2.1

Evaluate $\frac{x^{2}}{2}$ at $5$ and at $1$ .

$A (x) = \frac{1}{5 - 1} (6 ((\frac{5^{2}}{2}) - \frac{1^{2}}{2}))$

Step 7.2.2

Simplify.

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Step 7.2.2.1

Raise $5$ to the power of $2$ .

$A (x) = \frac{1}{5 - 1} (6 (\frac{25}{2} - \frac{1^{2}}{2}))$

Step 7.2.2.2

One to any power is one.

$A (x) = \frac{1}{5 - 1} (6 (\frac{25}{2} - \frac{1}{2}))$

Step 7.2.2.3

Combine the numerators over the common denominator.

$A (x) = \frac{1}{5 - 1} (6 (\frac{25 - 1}{2}))$

Step 7.2.2.4

Subtract $1$ from $25$ .

$A (x) = \frac{1}{5 - 1} (6 (\frac{24}{2}))$

Step 7.2.2.5

Cancel the common factor of $24$ and $2$ .

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Step 7.2.2.5.1

Factor $2$ out of $24$ .

$A (x) = \frac{1}{5 - 1} (6 (\frac{2 \cdot 12}{2}))$

Step 7.2.2.5.2

Cancel the common factors.

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Step 7.2.2.5.2.1

Factor $2$ out of $2$ .

$A (x) = \frac{1}{5 - 1} (6 (\frac{2 \cdot 12}{2 (1)}))$

Step 7.2.2.5.2.2

Cancel the common factor.

$A (x) = \frac{1}{5 - 1} (6 (\frac{2 \cdot 12}{2 \cdot 1}))$

Step 7.2.2.5.2.3

Rewrite the expression.

$A (x) = \frac{1}{5 - 1} (6 (\frac{12}{1}))$

Step 7.2.2.5.2.4

Divide $12$ by $1$ .

$A (x) = \frac{1}{5 - 1} (6 \cdot 12)$

Step 7.2.2.6

Multiply $6$ by $12$ .

$A (x) = \frac{1}{5 - 1} (72)$

Step 8

Subtract $1$ from $5$ .

$A (x) = \frac{1}{4} \cdot 72$

Step 9

Cancel the common factor of $4$ .

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Step 9.1

Factor $4$ out of $72$ .

$A (x) = \frac{1}{4} \cdot (4 (18))$

Step 9.2

Cancel the common factor.

$A (x) = \frac{1}{4} \cdot (4 \cdot 18)$

Step 9.3

Rewrite the expression.

$A (x) = 18$

Step 10