Calculus Examples

$y = 8 x (20 - x)$ ; $[0, 20]$

Step 1

Write $y = 8 x (20 - x)$ as a function.

$f (x) = 8 x (20 - x)$

Step 2

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 3

$f (x)$ is continuous on $[0, 20]$ .

$f (x)$ is continuous

Step 4

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 5

Substitute the actual values into the formula for the average value of a function.

$A (x) = \frac{1}{20 - 0} (\int_{0}^{20} 8 x (20 - x) d x)$

Step 6

Multiply $8 x (20 - x)$ .

$A (x) = \frac{1}{20 - 0} (\int_{0}^{20} 8 x \cdot 20 + 8 x (- x) d x)$

Step 7

Simplify.

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Step 7.1

Multiply $20$ by $8$ .

$A (x) = \frac{1}{20 - 0} (\int_{0}^{20} 160 x + 8 x (- x) d x)$

Step 7.2

Multiply $- 1$ by $8$ .

$A (x) = \frac{1}{20 - 0} (\int_{0}^{20} 160 x - 8 x \cdot x d x)$

Step 7.3

Raise $x$ to the power of $1$ .

$A (x) = \frac{1}{20 - 0} (\int_{0}^{20} 160 x - 8 (x \cdot x) d x)$

Step 7.4

Raise $x$ to the power of $1$ .

$A (x) = \frac{1}{20 - 0} (\int_{0}^{20} 160 x - 8 (x \cdot x) d x)$

Step 7.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$A (x) = \frac{1}{20 - 0} (\int_{0}^{20} 160 x - 8 x^{1 + 1} d x)$

Step 7.6

Add $1$ and $1$ .

$A (x) = \frac{1}{20 - 0} (\int_{0}^{20} 160 x - 8 x^{2} d x)$

Step 8

Split the single integral into multiple integrals.

$A (x) = \frac{1}{20 - 0} (\int_{0}^{20} 160 x d x + \int_{0}^{20} - 8 x^{2} d x)$

Step 9

Since $160$ is constant with respect to $x$ , move $160$ out of the integral.

$A (x) = \frac{1}{20 - 0} (160 \int_{0}^{20} x d x + \int_{0}^{20} - 8 x^{2} d x)$

Step 10

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

$A (x) = \frac{1}{20 - 0} (160 (\frac{1}{2} x^{2}]_{0}^{20}) + \int_{0}^{20} - 8 x^{2} d x)$

Step 11

Combine $\frac{1}{2}$ and $x^{2}$ .

$A (x) = \frac{1}{20 - 0} (160 (\frac{x^{2}}{2}]_{0}^{20}) + \int_{0}^{20} - 8 x^{2} d x)$

Step 12

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

$A (x) = \frac{1}{20 - 0} (160 (\frac{x^{2}}{2}]_{0}^{20}) - 8 \int_{0}^{20} x^{2} d x)$

Step 13

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

$A (x) = \frac{1}{20 - 0} (160 (\frac{x^{2}}{2}]_{0}^{20}) - 8 (\frac{1}{3} x^{3}]_{0}^{20}))$

Step 14

Simplify the answer.

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Step 14.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$A (x) = \frac{1}{20 - 0} (160 (\frac{x^{2}}{2}]_{0}^{20}) - 8 (\frac{x^{3}}{3}]_{0}^{20}))$

Step 14.2

Substitute and simplify.

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Step 14.2.1

Evaluate $\frac{x^{2}}{2}$ at $20$ and at $0$ .

$A (x) = \frac{1}{20 - 0} (160 ((\frac{20^{2}}{2}) - \frac{0^{2}}{2}) - 8 (\frac{x^{3}}{3}]_{0}^{20}))$

Step 14.2.2

Evaluate $\frac{x^{3}}{3}$ at $20$ and at $0$ .

$A (x) = \frac{1}{20 - 0} (160 (\frac{20^{2}}{2} - \frac{0^{2}}{2}) - 8 (\frac{20^{3}}{3} - \frac{0^{3}}{3}))$

Step 14.2.3

Simplify.

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Step 14.2.3.1

Raise $20$ to the power of $2$ .

$A (x) = \frac{1}{20 - 0} (160 (\frac{400}{2} - \frac{0^{2}}{2}) - 8 (\frac{20^{3}}{3} - \frac{0^{3}}{3}))$

Step 14.2.3.2

Cancel the common factor of $400$ and $2$ .

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Step 14.2.3.2.1

Factor $2$ out of $400$ .

$A (x) = \frac{1}{20 - 0} (160 (\frac{2 \cdot 200}{2} - \frac{0^{2}}{2}) - 8 (\frac{20^{3}}{3} - \frac{0^{3}}{3}))$

Step 14.2.3.2.2

Cancel the common factors.

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Step 14.2.3.2.2.1

Factor $2$ out of $2$ .

$A (x) = \frac{1}{20 - 0} (160 (\frac{2 \cdot 200}{2 (1)} - \frac{0^{2}}{2}) - 8 (\frac{20^{3}}{3} - \frac{0^{3}}{3}))$

Step 14.2.3.2.2.2

Cancel the common factor.

$A (x) = \frac{1}{20 - 0} (160 (\frac{2 \cdot 200}{2 \cdot 1} - \frac{0^{2}}{2}) - 8 (\frac{20^{3}}{3} - \frac{0^{3}}{3}))$

Step 14.2.3.2.2.3

Rewrite the expression.

$A (x) = \frac{1}{20 - 0} (160 (\frac{200}{1} - \frac{0^{2}}{2}) - 8 (\frac{20^{3}}{3} - \frac{0^{3}}{3}))$

Step 14.2.3.2.2.4

Divide $200$ by $1$ .

$A (x) = \frac{1}{20 - 0} (160 (200 - \frac{0^{2}}{2}) - 8 (\frac{20^{3}}{3} - \frac{0^{3}}{3}))$

Step 14.2.3.3

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

$A (x) = \frac{1}{20 - 0} (160 (200 - \frac{0}{2}) - 8 (\frac{20^{3}}{3} - \frac{0^{3}}{3}))$

Step 14.2.3.4

Cancel the common factor of $0$ and $2$ .

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Step 14.2.3.4.1

Factor $2$ out of $0$ .

$A (x) = \frac{1}{20 - 0} (160 (200 - \frac{2 (0)}{2}) - 8 (\frac{20^{3}}{3} - \frac{0^{3}}{3}))$

Step 14.2.3.4.2

Cancel the common factors.

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Step 14.2.3.4.2.1

Factor $2$ out of $2$ .

$A (x) = \frac{1}{20 - 0} (160 (200 - \frac{2 \cdot 0}{2 \cdot 1}) - 8 (\frac{20^{3}}{3} - \frac{0^{3}}{3}))$

Step 14.2.3.4.2.2

Cancel the common factor.

$A (x) = \frac{1}{20 - 0} (160 (200 - \frac{2 \cdot 0}{2 \cdot 1}) - 8 (\frac{20^{3}}{3} - \frac{0^{3}}{3}))$

Step 14.2.3.4.2.3

Rewrite the expression.

$A (x) = \frac{1}{20 - 0} (160 (200 - \frac{0}{1}) - 8 (\frac{20^{3}}{3} - \frac{0^{3}}{3}))$

Step 14.2.3.4.2.4

Divide $0$ by $1$ .

$A (x) = \frac{1}{20 - 0} (160 (200 - 0) - 8 (\frac{20^{3}}{3} - \frac{0^{3}}{3}))$

Step 14.2.3.5

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{20 - 0} (160 (200 + 0) - 8 (\frac{20^{3}}{3} - \frac{0^{3}}{3}))$

Step 14.2.3.6

Add $200$ and $0$ .

$A (x) = \frac{1}{20 - 0} (160 \cdot 200 - 8 (\frac{20^{3}}{3} - \frac{0^{3}}{3}))$

Step 14.2.3.7

Multiply $160$ by $200$ .

$A (x) = \frac{1}{20 - 0} (32000 - 8 (\frac{20^{3}}{3} - \frac{0^{3}}{3}))$

Step 14.2.3.8

Raise $20$ to the power of $3$ .

$A (x) = \frac{1}{20 - 0} (32000 - 8 (\frac{8000}{3} - \frac{0^{3}}{3}))$

Step 14.2.3.9

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

$A (x) = \frac{1}{20 - 0} (32000 - 8 (\frac{8000}{3} - \frac{0}{3}))$

Step 14.2.3.10

Cancel the common factor of $0$ and $3$ .

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Step 14.2.3.10.1

Factor $3$ out of $0$ .

$A (x) = \frac{1}{20 - 0} (32000 - 8 (\frac{8000}{3} - \frac{3 (0)}{3}))$

Step 14.2.3.10.2

Cancel the common factors.

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Step 14.2.3.10.2.1

Factor $3$ out of $3$ .

$A (x) = \frac{1}{20 - 0} (32000 - 8 (\frac{8000}{3} - \frac{3 \cdot 0}{3 \cdot 1}))$

Step 14.2.3.10.2.2

Cancel the common factor.

$A (x) = \frac{1}{20 - 0} (32000 - 8 (\frac{8000}{3} - \frac{3 \cdot 0}{3 \cdot 1}))$

Step 14.2.3.10.2.3

Rewrite the expression.

$A (x) = \frac{1}{20 - 0} (32000 - 8 (\frac{8000}{3} - \frac{0}{1}))$

Step 14.2.3.10.2.4

Divide $0$ by $1$ .

$A (x) = \frac{1}{20 - 0} (32000 - 8 (\frac{8000}{3} - 0))$

Step 14.2.3.11

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{20 - 0} (32000 - 8 (\frac{8000}{3} + 0))$

Step 14.2.3.12

Add $\frac{8000}{3}$ and $0$ .

$A (x) = \frac{1}{20 - 0} (32000 - 8 (\frac{8000}{3}))$

Step 14.2.3.13

Combine $- 8$ and $\frac{8000}{3}$ .

$A (x) = \frac{1}{20 - 0} (32000 + \frac{- 8 \cdot 8000}{3})$

Step 14.2.3.14

Multiply $- 8$ by $8000$ .

$A (x) = \frac{1}{20 - 0} (32000 + \frac{- 64000}{3})$

Step 14.2.3.15

Move the negative in front of the fraction.

$A (x) = \frac{1}{20 - 0} (32000 - \frac{64000}{3})$

Step 14.2.3.16

To write $32000$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$A (x) = \frac{1}{20 - 0} (32000 \cdot \frac{3}{3} - \frac{64000}{3})$

Step 14.2.3.17

Combine $32000$ and $\frac{3}{3}$ .

$A (x) = \frac{1}{20 - 0} (\frac{32000 \cdot 3}{3} - \frac{64000}{3})$

Step 14.2.3.18

Combine the numerators over the common denominator.

$A (x) = \frac{1}{20 - 0} (\frac{32000 \cdot 3 - 64000}{3})$

Step 14.2.3.19

Simplify the numerator.

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Step 14.2.3.19.1

Multiply $32000$ by $3$ .

$A (x) = \frac{1}{20 - 0} (\frac{96000 - 64000}{3})$

Step 14.2.3.19.2

Subtract $64000$ from $96000$ .

$A (x) = \frac{1}{20 - 0} (\frac{32000}{3})$

Step 15

Simplify the denominator.

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Step 15.1

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{20 + 0} \cdot \frac{32000}{3}$

Step 15.2

Add $20$ and $0$ .

$A (x) = \frac{1}{20} \cdot \frac{32000}{3}$

Step 16

Cancel the common factor of $20$ .

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Step 16.1

Factor $20$ out of $32000$ .

$A (x) = \frac{1}{20} \cdot \frac{20 (1600)}{3}$

Step 16.2

Cancel the common factor.

$A (x) = \frac{1}{20} \cdot \frac{20 \cdot 1600}{3}$

Step 16.3

Rewrite the expression.

$A (x) = \frac{1600}{3}$

Step 17