Calculus Examples

$f (x) = 6 \sqrt{x}$ , $[4, 9]$

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 $[4, 9]$ or not, find the domain of $f (x) = 6 \sqrt{x}$ .

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Step 1.1

Set the radicand in $\sqrt{x}$ greater than or equal to $0$ to find where the expression is defined.

$x \geq 0$

Step 1.2

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

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Interval Notation:

$[0, \infty)$

Set-Builder Notation:

${x | x \geq 0}$

Step 2

$f (x)$ is continuous on $[4, 9]$ .

$f (x)$ 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.

$A (x) = \frac{1}{9 - 4} (\int_{4}^{9} 6 \sqrt{x} d x)$

Step 5

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

$A (x) = \frac{1}{9 - 4} (6 \int_{4}^{9} \sqrt{x} d x)$

Step 6

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$A (x) = \frac{1}{9 - 4} (6 \int_{4}^{9} x^{\frac{1}{2}} d x)$

Step 7

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

$A (x) = \frac{1}{9 - 4} (6 (\frac{2}{3} x^{\frac{3}{2}}]_{4}^{9}))$

Step 8

Simplify the answer.

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Step 8.1

Combine $\frac{2}{3}$ and $x^{\frac{3}{2}}$ .

$A (x) = \frac{1}{9 - 4} (6 (\frac{2 x^{\frac{3}{2}}}{3}]_{4}^{9}))$

Step 8.2

Substitute and simplify.

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Step 8.2.1

Evaluate $\frac{2 x^{\frac{3}{2}}}{3}$ at $9$ and at $4$ .

$A (x) = \frac{1}{9 - 4} (6 ((\frac{2 \cdot 9^{\frac{3}{2}}}{3}) - \frac{2 \cdot 4^{\frac{3}{2}}}{3}))$

Step 8.2.2

Simplify.

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Step 8.2.2.1

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

$A (x) = \frac{1}{9 - 4} (6 (\frac{2 \cdot {(3^{2})}^{\frac{3}{2}}}{3} - \frac{2 \cdot 4^{\frac{3}{2}}}{3}))$

Step 8.2.2.2

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

$A (x) = \frac{1}{9 - 4} (6 (\frac{2 \cdot 3^{2 (\frac{3}{2})}}{3} - \frac{2 \cdot 4^{\frac{3}{2}}}{3}))$

Step 8.2.2.3

Cancel the common factor of $2$ .

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Step 8.2.2.3.1

Cancel the common factor.

$A (x) = \frac{1}{9 - 4} (6 (\frac{2 \cdot 3^{2 (\frac{3}{2})}}{3} - \frac{2 \cdot 4^{\frac{3}{2}}}{3}))$

Step 8.2.2.3.2

Rewrite the expression.

$A (x) = \frac{1}{9 - 4} (6 (\frac{2 \cdot 3^{3}}{3} - \frac{2 \cdot 4^{\frac{3}{2}}}{3}))$

Step 8.2.2.4

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

$A (x) = \frac{1}{9 - 4} (6 (\frac{2 \cdot 27}{3} - \frac{2 \cdot 4^{\frac{3}{2}}}{3}))$

Step 8.2.2.5

Multiply $2$ by $27$ .

$A (x) = \frac{1}{9 - 4} (6 (\frac{54}{3} - \frac{2 \cdot 4^{\frac{3}{2}}}{3}))$

Step 8.2.2.6

Cancel the common factor of $54$ and $3$ .

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Step 8.2.2.6.1

Factor $3$ out of $54$ .

$A (x) = \frac{1}{9 - 4} (6 (\frac{3 \cdot 18}{3} - \frac{2 \cdot 4^{\frac{3}{2}}}{3}))$

Step 8.2.2.6.2

Cancel the common factors.

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Step 8.2.2.6.2.1

Factor $3$ out of $3$ .

$A (x) = \frac{1}{9 - 4} (6 (\frac{3 \cdot 18}{3 (1)} - \frac{2 \cdot 4^{\frac{3}{2}}}{3}))$

Step 8.2.2.6.2.2

Cancel the common factor.

$A (x) = \frac{1}{9 - 4} (6 (\frac{3 \cdot 18}{3 \cdot 1} - \frac{2 \cdot 4^{\frac{3}{2}}}{3}))$

Step 8.2.2.6.2.3

Rewrite the expression.

$A (x) = \frac{1}{9 - 4} (6 (\frac{18}{1} - \frac{2 \cdot 4^{\frac{3}{2}}}{3}))$

Step 8.2.2.6.2.4

Divide $18$ by $1$ .

$A (x) = \frac{1}{9 - 4} (6 (18 - \frac{2 \cdot 4^{\frac{3}{2}}}{3}))$

Step 8.2.2.7

Rewrite $4$ as $2^{2}$ .

$A (x) = \frac{1}{9 - 4} (6 (18 - \frac{2 \cdot {(2^{2})}^{\frac{3}{2}}}{3}))$

Step 8.2.2.8

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

$A (x) = \frac{1}{9 - 4} (6 (18 - \frac{2 \cdot 2^{2 (\frac{3}{2})}}{3}))$

Step 8.2.2.9

Cancel the common factor of $2$ .

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Step 8.2.2.9.1

Cancel the common factor.

$A (x) = \frac{1}{9 - 4} (6 (18 - \frac{2 \cdot 2^{2 (\frac{3}{2})}}{3}))$

Step 8.2.2.9.2

Rewrite the expression.

$A (x) = \frac{1}{9 - 4} (6 (18 - \frac{2 \cdot 2^{3}}{3}))$

Step 8.2.2.10

Raise $2$ to the power of $3$ .

$A (x) = \frac{1}{9 - 4} (6 (18 - \frac{2 \cdot 8}{3}))$

Step 8.2.2.11

Multiply $2$ by $8$ .

$A (x) = \frac{1}{9 - 4} (6 (18 - \frac{16}{3}))$

Step 8.2.2.12

To write $18$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$A (x) = \frac{1}{9 - 4} (6 (18 \cdot \frac{3}{3} - \frac{16}{3}))$

Step 8.2.2.13

Combine $18$ and $\frac{3}{3}$ .

$A (x) = \frac{1}{9 - 4} (6 (\frac{18 \cdot 3}{3} - \frac{16}{3}))$

Step 8.2.2.14

Combine the numerators over the common denominator.

$A (x) = \frac{1}{9 - 4} (6 (\frac{18 \cdot 3 - 16}{3}))$

Step 8.2.2.15

Simplify the numerator.

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Step 8.2.2.15.1

Multiply $18$ by $3$ .

$A (x) = \frac{1}{9 - 4} (6 (\frac{54 - 16}{3}))$

Step 8.2.2.15.2

Subtract $16$ from $54$ .

$A (x) = \frac{1}{9 - 4} (6 (\frac{38}{3}))$

Step 8.2.2.16

Combine $6$ and $\frac{38}{3}$ .

$A (x) = \frac{1}{9 - 4} (\frac{6 \cdot 38}{3})$

Step 8.2.2.17

Multiply $6$ by $38$ .

$A (x) = \frac{1}{9 - 4} (\frac{228}{3})$

Step 8.2.2.18

Cancel the common factor of $228$ and $3$ .

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Step 8.2.2.18.1

Factor $3$ out of $228$ .

$A (x) = \frac{1}{9 - 4} (\frac{3 \cdot 76}{3})$

Step 8.2.2.18.2

Cancel the common factors.

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Step 8.2.2.18.2.1

Factor $3$ out of $3$ .

$A (x) = \frac{1}{9 - 4} (\frac{3 \cdot 76}{3 (1)})$

Step 8.2.2.18.2.2

Cancel the common factor.

$A (x) = \frac{1}{9 - 4} (\frac{3 \cdot 76}{3 \cdot 1})$

Step 8.2.2.18.2.3

Rewrite the expression.

$A (x) = \frac{1}{9 - 4} (\frac{76}{1})$

Step 8.2.2.18.2.4

Divide $76$ by $1$ .

$A (x) = \frac{1}{9 - 4} (76)$

Step 9

Subtract $4$ from $9$ .

$A (x) = \frac{1}{5} \cdot 76$

Step 10

Combine $\frac{1}{5}$ and $76$ .

$A (x) = \frac{76}{5}$

Step 11