Calculus Examples

$f (x) = \cos (x)$ , $[0, \frac{π}{6}]$

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 $[0, \frac{π}{6}]$ .

$f (x)$ 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.

$A (x) = \frac{1}{\frac{π}{6} - 0} (\int_{0}^{\frac{π}{6}} \cos (x) d x)$

Step 5

The integral of $\cos (x)$ with respect to $x$ is $\sin (x)$ .

$A (x) = \frac{1}{\frac{π}{6} - 0} (\sin (x)]_{0}^{\frac{π}{6}})$

Step 6

Simplify the answer.

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Step 6.1

Evaluate $\sin (x)$ at $\frac{π}{6}$ and at $0$ .

$A (x) = \frac{1}{\frac{π}{6} - 0} (\sin (\frac{π}{6}) - \sin (0))$

Step 6.2

Simplify.

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Step 6.2.1

The exact value of $\sin (\frac{π}{6})$ is $\frac{1}{2}$ .

$A (x) = \frac{1}{\frac{π}{6} - 0} (\frac{1}{2} - \sin (0))$

Step 6.2.2

The exact value of $\sin (0)$ is $0$ .

$A (x) = \frac{1}{\frac{π}{6} - 0} (\frac{1}{2} - 0)$

Step 6.2.3

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{\frac{π}{6} - 0} (\frac{1}{2} + 0)$

Step 6.2.4

Add $\frac{1}{2}$ and $0$ .

$A (x) = \frac{1}{\frac{π}{6} - 0} (\frac{1}{2})$

Step 7

Simplify the denominator.

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Step 7.1

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{\frac{π}{6} + 0} \cdot \frac{1}{2}$

Step 7.2

Add $\frac{π}{6}$ and $0$ .

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

Step 8

Multiply the numerator by the reciprocal of the denominator.

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

Step 9

Multiply $\frac{6}{π}$ by $1$ .

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

Step 10

Cancel the common factor of $2$ .

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Step 10.1

Factor $2$ out of $6$ .

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

Step 10.2

Cancel the common factor.

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

Step 10.3

Rewrite the expression.

$A (x) = \frac{3}{π}$

Step 11