Calculus Examples

$g (x) = x^{2} \sqrt{1 + x^{3}}$ , $[0, 2]$

Step 1

To find the average value of a function, the function should be continuous on the closed interval $[a, b]$ . To find whether $g (x)$ is continuous on $[0, 2]$ or not, find the domain of $g (x) = x^{2} \sqrt{1 + x^{3}}$ .

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Step 1.1

Set the radicand in $\sqrt{1 + x^{3}}$ greater than or equal to $0$ to find where the expression is defined.

$1 + x^{3} \geq 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Subtract $1$ from both sides of the inequality.

$x^{3} \geq - 1$

Step 1.2.2

Add $1$ to both sides of the inequality.

$x^{3} + 1 \geq 0$

Step 1.2.3

Convert the inequality to an equation.

$x^{3} + 1 = 0$

Step 1.2.4

Factor the left side of the equation.

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Step 1.2.4.1

Rewrite $1$ as $1^{3}$ .

$x^{3} + 1^{3} = 0$

Step 1.2.4.2

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = x$ and $b = 1$ .

$(x + 1) (x^{2} - x \cdot 1 + 1^{2}) = 0$

Step 1.2.4.3

Simplify.

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Step 1.2.4.3.1

Multiply $- 1$ by $1$ .

$(x + 1) (x^{2} - x + 1^{2}) = 0$

Step 1.2.4.3.2

One to any power is one.

$(x + 1) (x^{2} - x + 1) = 0$

Step 1.2.5

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 1 = 0$

$x^{2} - x + 1 = 0$

Step 1.2.6

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 1.2.6.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 1.2.6.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 1.2.7

Set $x^{2} - x + 1$ equal to $0$ and solve for $x$ .

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Step 1.2.7.1

Set $x^{2} - x + 1$ equal to $0$ .

$x^{2} - x + 1 = 0$

Step 1.2.7.2

Solve $x^{2} - x + 1 = 0$ for $x$ .

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Step 1.2.7.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 1.2.7.2.2

Substitute the values $a = 1$ , $b = - 1$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (1 \cdot 1)}}{2 \cdot 1}$

Step 1.2.7.2.3

Simplify.

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Step 1.2.7.2.3.1

Simplify the numerator.

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Step 1.2.7.2.3.1.1

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 1.2.7.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 1.2.7.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

Step 1.2.7.2.3.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

Step 1.2.7.2.3.1.3

Subtract $4$ from $1$ .

$x = \frac{1 \pm \sqrt{- 3}}{2 \cdot 1}$

Step 1.2.7.2.3.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{1 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Step 1.2.7.2.3.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Step 1.2.7.2.3.1.6

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

$x = \frac{1 \pm i \sqrt{3}}{2 \cdot 1}$

Step 1.2.7.2.3.2

Multiply $2$ by $1$ .

$x = \frac{1 \pm i \sqrt{3}}{2}$

Step 1.2.7.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 1.2.7.2.4.1

Simplify the numerator.

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Step 1.2.7.2.4.1.1

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 1.2.7.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 1.2.7.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

Step 1.2.7.2.4.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

Step 1.2.7.2.4.1.3

Subtract $4$ from $1$ .

$x = \frac{1 \pm \sqrt{- 3}}{2 \cdot 1}$

Step 1.2.7.2.4.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{1 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Step 1.2.7.2.4.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Step 1.2.7.2.4.1.6

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

$x = \frac{1 \pm i \sqrt{3}}{2 \cdot 1}$

Step 1.2.7.2.4.2

Multiply $2$ by $1$ .

$x = \frac{1 \pm i \sqrt{3}}{2}$

Step 1.2.7.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{1 + i \sqrt{3}}{2}$

Step 1.2.7.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 1.2.7.2.5.1

Simplify the numerator.

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Step 1.2.7.2.5.1.1

Raise $- 1$ to the power of $2$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

Step 1.2.7.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 1$ .

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Step 1.2.7.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4 \cdot 1}}{2 \cdot 1}$

Step 1.2.7.2.5.1.2.2

Multiply $- 4$ by $1$ .

$x = \frac{1 \pm \sqrt{1 - 4}}{2 \cdot 1}$

Step 1.2.7.2.5.1.3

Subtract $4$ from $1$ .

$x = \frac{1 \pm \sqrt{- 3}}{2 \cdot 1}$

Step 1.2.7.2.5.1.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{1 \pm \sqrt{- 1 \cdot 3}}{2 \cdot 1}$

Step 1.2.7.2.5.1.5

Rewrite $\sqrt{- 1 (3)}$ as $\sqrt{- 1} \cdot \sqrt{3}$ .

$x = \frac{1 \pm \sqrt{- 1} \cdot \sqrt{3}}{2 \cdot 1}$

Step 1.2.7.2.5.1.6

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

$x = \frac{1 \pm i \sqrt{3}}{2 \cdot 1}$

Step 1.2.7.2.5.2

Multiply $2$ by $1$ .

$x = \frac{1 \pm i \sqrt{3}}{2}$

Step 1.2.7.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{1 - i \sqrt{3}}{2}$

Step 1.2.7.2.6

The final answer is the combination of both solutions.

$x = \frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2}$

Step 1.2.8

The final solution is all the values that make $(x + 1) (x^{2} - x + 1) = 0$ true.

$x = - 1, \frac{1 + i \sqrt{3}}{2}, \frac{1 - i \sqrt{3}}{2}$

Step 1.2.9

The solution consists of all of the true intervals.

$x \geq - 1$

Step 1.3

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

Interval Notation:

$[- 1, \infty)$

Set-Builder Notation:

${x | x \geq - 1}$

Interval Notation:

$[- 1, \infty)$

Set-Builder Notation:

${x | x \geq - 1}$

Step 2

$g (x)$ is continuous on $[0, 2]$ .

$g (x)$ is continuous

Step 3

The average value of function $g$ 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.

$A (x) = \frac{1}{2 - 0} (\int_{0}^{2} x^{2} \sqrt{1 + x^{3}} d x)$

Step 5

Let $u = 1 + x^{3}$ . Then $d u = 3 x^{2} d x$ , so $\frac{1}{3} d u = x^{2} d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = 1 + x^{3}$ . Find $\frac{d u}{d x}$ .

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Step 5.1.1

Differentiate $1 + x^{3}$ .

$\frac{d}{d x} [1 + x^{3}]$

Step 5.1.2

By the Sum Rule, the derivative of $1 + x^{3}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [x^{3}]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [x^{3}]$

Step 5.1.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [x^{3}]$

Step 5.1.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$0 + 3 x^{2}$

Step 5.1.5

Add $0$ and $3 x^{2}$ .

$3 x^{2}$

Step 5.2

Substitute the lower limit in for $x$ in $u = 1 + x^{3}$ .

$u_{lower} = 1 + 0^{3}$

Step 5.3

Simplify.

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Step 5.3.1

Raising $0$ to any positive power yields $0$ .

$u_{lower} = 1 + 0$

Step 5.3.2

Add $1$ and $0$ .

$u_{lower} = 1$

Step 5.4

Substitute the upper limit in for $x$ in $u = 1 + x^{3}$ .

$u_{upper} = 1 + 2^{3}$

Step 5.5

Simplify.

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Step 5.5.1

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

$u_{upper} = 1 + 8$

Step 5.5.2

Add $1$ and $8$ .

$u_{upper} = 9$

Step 5.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 1$

$u_{upper} = 9$

Step 5.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$A (x) = \frac{1}{2 - 0} (\int_{1}^{9} \sqrt{u} (\frac{1}{3}) d u)$

Step 6

Combine $\sqrt{u}$ and $\frac{1}{3}$ .

$A (x) = \frac{1}{2 - 0} (\int_{1}^{9} \frac{\sqrt{u}}{3} d u)$

Step 7

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$A (x) = \frac{1}{2 - 0} (\frac{1}{3} \cdot \int_{1}^{9} \sqrt{u} d u)$

Step 8

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

$A (x) = \frac{1}{2 - 0} (\frac{1}{3} \cdot \int_{1}^{9} u^{\frac{1}{2}} d u)$

Step 9

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

$A (x) = \frac{1}{2 - 0} (\frac{1}{3} \cdot \frac{2}{3} u^{\frac{3}{2}}]_{1}^{9})$

Step 10

Substitute and simplify.

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Step 10.1

Evaluate $\frac{2}{3} u^{\frac{3}{2}}$ at $9$ and at $1$ .

$A (x) = \frac{1}{2 - 0} (\frac{1}{3} \cdot ((\frac{2}{3} \cdot 9^{\frac{3}{2}}) - \frac{2}{3} \cdot 1^{\frac{3}{2}}))$

Step 10.2

Simplify.

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Step 10.2.1

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

$A (x) = \frac{1}{2 - 0} (\frac{1}{3} \cdot (\frac{2}{3} \cdot {(3^{2})}^{\frac{3}{2}} - \frac{2}{3} \cdot 1^{\frac{3}{2}}))$

Step 10.2.2

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

$A (x) = \frac{1}{2 - 0} (\frac{1}{3} \cdot (\frac{2}{3} \cdot 3^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 1^{\frac{3}{2}}))$

Step 10.2.3

Cancel the common factor of $2$ .

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Step 10.2.3.1

Cancel the common factor.

$A (x) = \frac{1}{2 - 0} (\frac{1}{3} \cdot (\frac{2}{3} \cdot 3^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 1^{\frac{3}{2}}))$

Step 10.2.3.2

Rewrite the expression.

$A (x) = \frac{1}{2 - 0} (\frac{1}{3} \cdot (\frac{2}{3} \cdot 3^{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}}))$

Step 10.2.4

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

$A (x) = \frac{1}{2 - 0} (\frac{1}{3} \cdot (\frac{2}{3} \cdot 27 - \frac{2}{3} \cdot 1^{\frac{3}{2}}))$

Step 10.2.5

Combine $\frac{2}{3}$ and $27$ .

$A (x) = \frac{1}{2 - 0} (\frac{1}{3} \cdot (\frac{2 \cdot 27}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}}))$

Step 10.2.6

Multiply $2$ by $27$ .

$A (x) = \frac{1}{2 - 0} (\frac{1}{3} \cdot (\frac{54}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}}))$

Step 10.2.7

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

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Step 10.2.7.1

Factor $3$ out of $54$ .

$A (x) = \frac{1}{2 - 0} (\frac{1}{3} \cdot (\frac{3 \cdot 18}{3} - \frac{2}{3} \cdot 1^{\frac{3}{2}}))$

Step 10.2.7.2

Cancel the common factors.

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Step 10.2.7.2.1

Factor $3$ out of $3$ .

$A (x) = \frac{1}{2 - 0} (\frac{1}{3} \cdot (\frac{3 \cdot 18}{3 (1)} - \frac{2}{3} \cdot 1^{\frac{3}{2}}))$

Step 10.2.7.2.2

Cancel the common factor.

$A (x) = \frac{1}{2 - 0} (\frac{1}{3} \cdot (\frac{3 \cdot 18}{3 \cdot 1} - \frac{2}{3} \cdot 1^{\frac{3}{2}}))$

Step 10.2.7.2.3

Rewrite the expression.

$A (x) = \frac{1}{2 - 0} (\frac{1}{3} \cdot (\frac{18}{1} - \frac{2}{3} \cdot 1^{\frac{3}{2}}))$

Step 10.2.7.2.4

Divide $18$ by $1$ .

$A (x) = \frac{1}{2 - 0} (\frac{1}{3} \cdot (18 - \frac{2}{3} \cdot 1^{\frac{3}{2}}))$

Step 10.2.8

One to any power is one.

$A (x) = \frac{1}{2 - 0} (\frac{1}{3} \cdot (18 - \frac{2}{3} \cdot 1))$

Step 10.2.9

Multiply $- 1$ by $1$ .

$A (x) = \frac{1}{2 - 0} (\frac{1}{3} \cdot (18 - \frac{2}{3}))$

Step 10.2.10

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

$A (x) = \frac{1}{2 - 0} (\frac{1}{3} \cdot (18 \cdot \frac{3}{3} - \frac{2}{3}))$

Step 10.2.11

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

$A (x) = \frac{1}{2 - 0} (\frac{1}{3} \cdot (\frac{18 \cdot 3}{3} - \frac{2}{3}))$

Step 10.2.12

Combine the numerators over the common denominator.

$A (x) = \frac{1}{2 - 0} (\frac{1}{3} \cdot \frac{18 \cdot 3 - 2}{3})$

Step 10.2.13

Simplify the numerator.

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Step 10.2.13.1

Multiply $18$ by $3$ .

$A (x) = \frac{1}{2 - 0} (\frac{1}{3} \cdot \frac{54 - 2}{3})$

Step 10.2.13.2

Subtract $2$ from $54$ .

$A (x) = \frac{1}{2 - 0} (\frac{1}{3} \cdot \frac{52}{3})$

Step 10.2.14

Multiply $\frac{1}{3}$ by $\frac{52}{3}$ .

$A (x) = \frac{1}{2 - 0} (\frac{52}{3 \cdot 3})$

Step 10.2.15

Multiply $3$ by $3$ .

$A (x) = \frac{1}{2 - 0} (\frac{52}{9})$

Step 11

Simplify the denominator.

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Step 11.1

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{2 + 0} \cdot \frac{52}{9}$

Step 11.2

Add $2$ and $0$ .

$A (x) = \frac{1}{2} \cdot \frac{52}{9}$

Step 12

Cancel the common factor of $2$ .

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Step 12.1

Factor $2$ out of $52$ .

$A (x) = \frac{1}{2} \cdot \frac{2 (26)}{9}$

Step 12.2

Cancel the common factor.

$A (x) = \frac{1}{2} \cdot \frac{2 \cdot 26}{9}$

Step 12.3

Rewrite the expression.

$A (x) = \frac{26}{9}$

Step 13