Calculus Examples

$f (x) = \frac{8}{x^{2}}$ , $[2, 4]$

Step 1

To find the average value of a function, the function should be continuous on the closed interval $[a, b]$ . To find whether $f (x)$ is continuous on $[2, 4]$ or not, find the domain of $f (x) = \frac{8}{x^{2}}$ .

Tap for more steps...

Step 1.1

Set the denominator in $\frac{8}{x^{2}}$ equal to $0$ to find where the expression is undefined.

$x^{2} = 0$

Step 1.2

Solve for $x$ .

Tap for more steps...

Step 1.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{0}$

Step 1.2.2

Simplify $\pm \sqrt{0}$ .

Tap for more steps...

Step 1.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 1.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 1.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 1.3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Step 2

$f (x)$ is continuous on $[2, 4]$ .

$f (x)$ 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.

$A (x) = \frac{1}{4 - 2} (\int_{2}^{4} \frac{8}{x^{2}} d x)$

Step 5

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$A (x) = \frac{1}{4 - 2} (8 \int_{2}^{4} \frac{1}{x^{2}} d x)$

Step 6

Apply basic rules of exponents.

Tap for more steps...

Step 6.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

$A (x) = \frac{1}{4 - 2} (8 \int_{2}^{4} {(x^{2})}^{- 1} d x)$

Step 6.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

Tap for more steps...

Step 6.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$A (x) = \frac{1}{4 - 2} (8 \int_{2}^{4} x^{2 \cdot - 1} d x)$

Step 6.2.2

Multiply $2$ by $- 1$ .

$A (x) = \frac{1}{4 - 2} (8 \int_{2}^{4} x^{- 2} d x)$

Step 7

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

$A (x) = \frac{1}{4 - 2} (8 (- x^{- 1}]_{2}^{4}))$

Step 8

Substitute and simplify.

Tap for more steps...

Step 8.1

Evaluate $- x^{- 1}$ at $4$ and at $2$ .

$A (x) = \frac{1}{4 - 2} (8 ((- 4^{- 1}) + 2^{- 1}))$

Step 8.2

Simplify.

Tap for more steps...

Step 8.2.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$A (x) = \frac{1}{4 - 2} (8 (- \frac{1}{4} + 2^{- 1}))$

Step 8.2.2

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$A (x) = \frac{1}{4 - 2} (8 (- \frac{1}{4} + \frac{1}{2}))$

Step 8.2.3

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$A (x) = \frac{1}{4 - 2} (8 (- \frac{1}{4} + \frac{1}{2} \cdot \frac{2}{2}))$

Step 8.2.4

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 8.2.4.1

Multiply $\frac{1}{2}$ by $\frac{2}{2}$ .

$A (x) = \frac{1}{4 - 2} (8 (- \frac{1}{4} + \frac{2}{2 \cdot 2}))$

Step 8.2.4.2

Multiply $2$ by $2$ .

$A (x) = \frac{1}{4 - 2} (8 (- \frac{1}{4} + \frac{2}{4}))$

Step 8.2.5

Combine the numerators over the common denominator.

$A (x) = \frac{1}{4 - 2} (8 (\frac{- 1 + 2}{4}))$

Step 8.2.6

Add $- 1$ and $2$ .

$A (x) = \frac{1}{4 - 2} (8 (\frac{1}{4}))$

Step 8.2.7

Combine $8$ and $\frac{1}{4}$ .

$A (x) = \frac{1}{4 - 2} (\frac{8}{4})$

Step 8.2.8

Cancel the common factor of $8$ and $4$ .

Tap for more steps...

Step 8.2.8.1

Factor $4$ out of $8$ .

$A (x) = \frac{1}{4 - 2} (\frac{4 \cdot 2}{4})$

Step 8.2.8.2

Cancel the common factors.

Tap for more steps...

Step 8.2.8.2.1

Factor $4$ out of $4$ .

$A (x) = \frac{1}{4 - 2} (\frac{4 \cdot 2}{4 (1)})$

Step 8.2.8.2.2

Cancel the common factor.

$A (x) = \frac{1}{4 - 2} (\frac{4 \cdot 2}{4 \cdot 1})$

Step 8.2.8.2.3

Rewrite the expression.

$A (x) = \frac{1}{4 - 2} (\frac{2}{1})$

Step 8.2.8.2.4

Divide $2$ by $1$ .

$A (x) = \frac{1}{4 - 2} (2)$

Step 9

Subtract $2$ from $4$ .

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

Step 10

Cancel the common factor of $2$ .

Tap for more steps...

Step 10.1

Cancel the common factor.

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

Step 10.2

Rewrite the expression.

$A (x) = 1$

Step 11