Calculus Examples

$f (x) = \frac{4}{x^{2} + 9}$ , $[- 3, 3]$

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 $[- 3, 3]$ or not, find the domain of $f (x) = \frac{4}{x^{2} + 9}$ .

Tap for more steps...

Step 1.1

Set the denominator in $\frac{4}{x^{2} + 9}$ equal to $0$ to find where the expression is undefined.

$x^{2} + 9 = 0$

Step 1.2

Solve for $x$ .

Tap for more steps...

Step 1.2.1

Subtract $9$ from both sides of the equation.

$x^{2} = - 9$

Step 1.2.2

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

$x = \pm \sqrt{- 9}$

Step 1.2.3

Simplify $\pm \sqrt{- 9}$ .

Tap for more steps...

Step 1.2.3.1

Rewrite $- 9$ as $- 1 (9)$ .

$x = \pm \sqrt{- 1 (9)}$

Step 1.2.3.2

Rewrite $\sqrt{- 1 (9)}$ as $\sqrt{- 1} \cdot \sqrt{9}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{9}$

Step 1.2.3.3

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \cdot \sqrt{9}$

Step 1.2.3.4

Rewrite $9$ as $3^{2}$ .

$x = \pm i \cdot \sqrt{3^{2}}$

Step 1.2.3.5

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

$x = \pm i \cdot 3$

Step 1.2.3.6

Move $3$ to the left of $i$ .

$x = \pm 3 i$

Step 1.2.4

The complete solution is the result of both the positive and negative portions of the solution.

Tap for more steps...

Step 1.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 3 i$

Step 1.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 3 i$

Step 1.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 3 i, - 3 i$

Step 1.3

The domain is all real numbers.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 2

$f (x)$ is continuous on $[- 3, 3]$ .

$f (x)$ 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.

$A (x) = \frac{1}{3 + 3} (\int_{- 3}^{3} \frac{4}{x^{2} + 9} d x)$

Step 5

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$A (x) = \frac{1}{3 + 3} (4 \int_{- 3}^{3} \frac{1}{x^{2} + 9} d x)$

Step 6

Simplify the expression.

Tap for more steps...

Step 6.1

Reorder $x^{2}$ and $9$ .

$A (x) = \frac{1}{3 + 3} (4 \int_{- 3}^{3} \frac{1}{9 + x^{2}} d x)$

Step 6.2

Rewrite $9$ as $3^{2}$ .

$A (x) = \frac{1}{3 + 3} (4 \int_{- 3}^{3} \frac{1}{3^{2} + x^{2}} d x)$

Step 7

The integral of $\frac{1}{3^{2} + x^{2}}$ with respect to $x$ is $\frac{1}{3} \arctan (\frac{x}{3})]_{- 3}^{3}$ .

$A (x) = \frac{1}{3 + 3} (4 (\frac{1}{3} \cdot \arctan (\frac{x}{3})]_{- 3}^{3}))$

Step 8

Simplify the answer.

Tap for more steps...

Step 8.1

Combine $\frac{1}{3}$ and $\arctan (\frac{x}{3})$ .

$A (x) = \frac{1}{3 + 3} (4 (\frac{\arctan (\frac{x}{3})}{3}]_{- 3}^{3}))$

Step 8.2

Substitute and simplify.

Tap for more steps...

Step 8.2.1

Evaluate $\frac{\arctan (\frac{x}{3})}{3}$ at $3$ and at $- 3$ .

$A (x) = \frac{1}{3 + 3} (4 ((\frac{\arctan (\frac{3}{3})}{3}) - \frac{\arctan (\frac{- 3}{3})}{3}))$

Step 8.2.2

Simplify.

Tap for more steps...

Step 8.2.2.1

Cancel the common factor of $3$ .

Tap for more steps...

Step 8.2.2.1.1

Cancel the common factor.

$A (x) = \frac{1}{3 + 3} (4 (\frac{\arctan (\frac{3}{3})}{3} - \frac{\arctan (\frac{- 3}{3})}{3}))$

Step 8.2.2.1.2

Rewrite the expression.

$A (x) = \frac{1}{3 + 3} (4 (\frac{\arctan (1)}{3} - \frac{\arctan (\frac{- 3}{3})}{3}))$

Step 8.2.2.2

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

Tap for more steps...

Step 8.2.2.2.1

Factor $3$ out of $- 3$ .

$A (x) = \frac{1}{3 + 3} (4 (\frac{\arctan (1)}{3} - \frac{\arctan (\frac{3 \cdot - 1}{3})}{3}))$

Step 8.2.2.2.2

Cancel the common factors.

Tap for more steps...

Step 8.2.2.2.2.1

Factor $3$ out of $3$ .

$A (x) = \frac{1}{3 + 3} (4 (\frac{\arctan (1)}{3} - \frac{\arctan (\frac{3 \cdot - 1}{3 (1)})}{3}))$

Step 8.2.2.2.2.2

Cancel the common factor.

$A (x) = \frac{1}{3 + 3} (4 (\frac{\arctan (1)}{3} - \frac{\arctan (\frac{3 \cdot - 1}{3 \cdot 1})}{3}))$

Step 8.2.2.2.2.3

Rewrite the expression.

$A (x) = \frac{1}{3 + 3} (4 (\frac{\arctan (1)}{3} - \frac{\arctan (\frac{- 1}{1})}{3}))$

Step 8.2.2.2.2.4

Divide $- 1$ by $1$ .

$A (x) = \frac{1}{3 + 3} (4 (\frac{\arctan (1)}{3} - \frac{\arctan (- 1)}{3}))$

Step 9

Simplify.

Tap for more steps...

Step 9.1

Combine the numerators over the common denominator.

$A (x) = \frac{1}{3 + 3} (4 (\frac{\arctan (1) - \arctan (- 1)}{3}))$

Step 9.2

Simplify each term.

Tap for more steps...

Step 9.2.1

The exact value of $\arctan (1)$ is $\frac{π}{4}$ .

$A (x) = \frac{1}{3 + 3} (4 (\frac{\frac{π}{4} - \arctan (- 1)}{3}))$

Step 9.2.2

The exact value of $\arctan (- 1)$ is $- \frac{π}{4}$ .

$A (x) = \frac{1}{3 + 3} (4 (\frac{\frac{π}{4} + \frac{π}{4}}{3}))$

Step 9.2.3

Multiply $- - \frac{π}{4}$ .

Tap for more steps...

Step 9.2.3.1

Multiply $- 1$ by $- 1$ .

$A (x) = \frac{1}{3 + 3} (4 (\frac{\frac{π}{4} + 1 (\frac{π}{4})}{3}))$

Step 9.2.3.2

Multiply $\frac{π}{4}$ by $1$ .

$A (x) = \frac{1}{3 + 3} (4 (\frac{\frac{π}{4} + \frac{π}{4}}{3}))$

Step 9.3

Combine the numerators over the common denominator.

$A (x) = \frac{1}{3 + 3} (4 (\frac{\frac{π + π}{4}}{3}))$

Step 9.4

Add $π$ and $π$ .

$A (x) = \frac{1}{3 + 3} (4 (\frac{\frac{2 π}{4}}{3}))$

Step 9.5

Cancel the common factor of $2$ and $4$ .

Tap for more steps...

Step 9.5.1

Factor $2$ out of $2 π$ .

$A (x) = \frac{1}{3 + 3} (4 (\frac{\frac{2 (π)}{4}}{3}))$

Step 9.5.2

Cancel the common factors.

Tap for more steps...

Step 9.5.2.1

Factor $2$ out of $4$ .

$A (x) = \frac{1}{3 + 3} (4 (\frac{\frac{2 π}{2 \cdot 2}}{3}))$

Step 9.5.2.2

Cancel the common factor.

$A (x) = \frac{1}{3 + 3} (4 (\frac{\frac{2 π}{2 \cdot 2}}{3}))$

Step 9.5.2.3

Rewrite the expression.

$A (x) = \frac{1}{3 + 3} (4 (\frac{\frac{π}{2}}{3}))$

Step 9.6

Multiply the numerator by the reciprocal of the denominator.

$A (x) = \frac{1}{3 + 3} (4 (\frac{π}{2} \cdot \frac{1}{3}))$

Step 9.7

Multiply $\frac{π}{2} \cdot \frac{1}{3}$ .

Tap for more steps...

Step 9.7.1

Multiply $\frac{π}{2}$ by $\frac{1}{3}$ .

$A (x) = \frac{1}{3 + 3} (4 (\frac{π}{2 \cdot 3}))$

Step 9.7.2

Multiply $2$ by $3$ .

$A (x) = \frac{1}{3 + 3} (4 (\frac{π}{6}))$

Step 9.8

Cancel the common factor of $2$ .

Tap for more steps...

Step 9.8.1

Factor $2$ out of $4$ .

$A (x) = \frac{1}{3 + 3} (2 (2) (\frac{π}{6}))$

Step 9.8.2

Factor $2$ out of $6$ .

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

Step 9.8.3

Cancel the common factor.

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

Step 9.8.4

Rewrite the expression.

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

Step 9.9

Combine $2$ and $\frac{π}{3}$ .

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

Step 10

Add $3$ and $3$ .

$A (x) = \frac{1}{6} \cdot \frac{2 π}{3}$

Step 11

Cancel the common factor of $2$ .

Tap for more steps...

Step 11.1

Factor $2$ out of $6$ .

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

Step 11.2

Factor $2$ out of $2 π$ .

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

Step 11.3

Cancel the common factor.

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

Step 11.4

Rewrite the expression.

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

Step 12

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

$A (x) = \frac{π}{3 \cdot 3}$

Step 13

Multiply $3$ by $3$ .

$A (x) = \frac{π}{9}$

Step 14