Calculus Examples

$f (x) = 204 x - 2 x^{2} + 3 x^{2} + 14 x + 3$ , $[0, 20]$

Step 1

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 2

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

$f (x)$ 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.

$A (x) = \frac{1}{20 - 0} (\int_{0}^{20} 204 x - 2 x^{2} + 3 x^{2} + 14 x + 3 d x)$

Step 5

Simplify.

Tap for more steps...

Step 5.1

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

$A (x) = \frac{1}{20 - 0} (\int_{0}^{20} 204 x + x^{2} + 14 x + 3 d x)$

Step 5.2

Add $204 x$ and $14 x$ .

$A (x) = \frac{1}{20 - 0} (\int_{0}^{20} x^{2} + 218 x + 3 d x)$

Step 6

Split the single integral into multiple integrals.

$A (x) = \frac{1}{20 - 0} (\int_{0}^{20} x^{2} d x + \int_{0}^{20} 218 x d x + \int_{0}^{20} 3 d x)$

Step 7

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

$A (x) = \frac{1}{20 - 0} (\frac{1}{3} x^{3}]_{0}^{20} + \int_{0}^{20} 218 x d x + \int_{0}^{20} 3 d x)$

Step 8

Since $218$ is constant with respect to $x$ , move $218$ out of the integral.

$A (x) = \frac{1}{20 - 0} (\frac{1}{3} x^{3}]_{0}^{20} + 218 \int_{0}^{20} x d x + \int_{0}^{20} 3 d x)$

Step 9

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

$A (x) = \frac{1}{20 - 0} (\frac{1}{3} x^{3}]_{0}^{20} + 218 (\frac{1}{2} x^{2}]_{0}^{20}) + \int_{0}^{20} 3 d x)$

Step 10

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

$A (x) = \frac{1}{20 - 0} (\frac{1}{3} x^{3}]_{0}^{20} + 218 (\frac{x^{2}}{2}]_{0}^{20}) + \int_{0}^{20} 3 d x)$

Step 11

Apply the constant rule.

$A (x) = \frac{1}{20 - 0} (\frac{1}{3} x^{3}]_{0}^{20} + 218 (\frac{x^{2}}{2}]_{0}^{20}) + 3 x]_{0}^{20})$

Step 12

Simplify the answer.

Tap for more steps...

Step 12.1

Combine $\frac{1}{3} x^{3}]_{0}^{20}$ and $3 x]_{0}^{20}$ .

$A (x) = \frac{1}{20 - 0} (218 (\frac{x^{2}}{2}]_{0}^{20}) + \frac{1}{3} x^{3} + 3 x]_{0}^{20})$

Step 12.2

Substitute and simplify.

Tap for more steps...

Step 12.2.1

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

$A (x) = \frac{1}{20 - 0} (218 ((\frac{20^{2}}{2}) - \frac{0^{2}}{2}) + \frac{1}{3} x^{3} + 3 x]_{0}^{20})$

Step 12.2.2

Evaluate $\frac{1}{3} x^{3} + 3 x$ at $20$ and at $0$ .

$A (x) = \frac{1}{20 - 0} (218 ((\frac{20^{2}}{2}) - \frac{0^{2}}{2}) + \frac{1}{3} \cdot 20^{3} + 3 \cdot 20 - (\frac{1}{3} \cdot 0^{3} + 3 \cdot 0))$

Step 12.2.3

Simplify.

Tap for more steps...

Step 12.2.3.1

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

$A (x) = \frac{1}{20 - 0} (218 (\frac{400}{2} - \frac{0^{2}}{2}) + \frac{1}{3} \cdot 20^{3} + 3 \cdot 20 - (\frac{1}{3} \cdot 0^{3} + 3 \cdot 0))$

Step 12.2.3.2

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

Tap for more steps...

Step 12.2.3.2.1

Factor $2$ out of $400$ .

$A (x) = \frac{1}{20 - 0} (218 (\frac{2 \cdot 200}{2} - \frac{0^{2}}{2}) + \frac{1}{3} \cdot 20^{3} + 3 \cdot 20 - (\frac{1}{3} \cdot 0^{3} + 3 \cdot 0))$

Step 12.2.3.2.2

Cancel the common factors.

Tap for more steps...

Step 12.2.3.2.2.1

Factor $2$ out of $2$ .

$A (x) = \frac{1}{20 - 0} (218 (\frac{2 \cdot 200}{2 (1)} - \frac{0^{2}}{2}) + \frac{1}{3} \cdot 20^{3} + 3 \cdot 20 - (\frac{1}{3} \cdot 0^{3} + 3 \cdot 0))$

Step 12.2.3.2.2.2

Cancel the common factor.

$A (x) = \frac{1}{20 - 0} (218 (\frac{2 \cdot 200}{2 \cdot 1} - \frac{0^{2}}{2}) + \frac{1}{3} \cdot 20^{3} + 3 \cdot 20 - (\frac{1}{3} \cdot 0^{3} + 3 \cdot 0))$

Step 12.2.3.2.2.3

Rewrite the expression.

$A (x) = \frac{1}{20 - 0} (218 (\frac{200}{1} - \frac{0^{2}}{2}) + \frac{1}{3} \cdot 20^{3} + 3 \cdot 20 - (\frac{1}{3} \cdot 0^{3} + 3 \cdot 0))$

Step 12.2.3.2.2.4

Divide $200$ by $1$ .

$A (x) = \frac{1}{20 - 0} (218 (200 - \frac{0^{2}}{2}) + \frac{1}{3} \cdot 20^{3} + 3 \cdot 20 - (\frac{1}{3} \cdot 0^{3} + 3 \cdot 0))$

Step 12.2.3.3

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

$A (x) = \frac{1}{20 - 0} (218 (200 - \frac{0}{2}) + \frac{1}{3} \cdot 20^{3} + 3 \cdot 20 - (\frac{1}{3} \cdot 0^{3} + 3 \cdot 0))$

Step 12.2.3.4

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

Tap for more steps...

Step 12.2.3.4.1

Factor $2$ out of $0$ .

$A (x) = \frac{1}{20 - 0} (218 (200 - \frac{2 (0)}{2}) + \frac{1}{3} \cdot 20^{3} + 3 \cdot 20 - (\frac{1}{3} \cdot 0^{3} + 3 \cdot 0))$

Step 12.2.3.4.2

Cancel the common factors.

Tap for more steps...

Step 12.2.3.4.2.1

Factor $2$ out of $2$ .

$A (x) = \frac{1}{20 - 0} (218 (200 - \frac{2 \cdot 0}{2 \cdot 1}) + \frac{1}{3} \cdot 20^{3} + 3 \cdot 20 - (\frac{1}{3} \cdot 0^{3} + 3 \cdot 0))$

Step 12.2.3.4.2.2

Cancel the common factor.

$A (x) = \frac{1}{20 - 0} (218 (200 - \frac{2 \cdot 0}{2 \cdot 1}) + \frac{1}{3} \cdot 20^{3} + 3 \cdot 20 - (\frac{1}{3} \cdot 0^{3} + 3 \cdot 0))$

Step 12.2.3.4.2.3

Rewrite the expression.

$A (x) = \frac{1}{20 - 0} (218 (200 - \frac{0}{1}) + \frac{1}{3} \cdot 20^{3} + 3 \cdot 20 - (\frac{1}{3} \cdot 0^{3} + 3 \cdot 0))$

Step 12.2.3.4.2.4

Divide $0$ by $1$ .

$A (x) = \frac{1}{20 - 0} (218 (200 - 0) + \frac{1}{3} \cdot 20^{3} + 3 \cdot 20 - (\frac{1}{3} \cdot 0^{3} + 3 \cdot 0))$

Step 12.2.3.5

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{20 - 0} (218 (200 + 0) + \frac{1}{3} \cdot 20^{3} + 3 \cdot 20 - (\frac{1}{3} \cdot 0^{3} + 3 \cdot 0))$

Step 12.2.3.6

Add $200$ and $0$ .

$A (x) = \frac{1}{20 - 0} (218 \cdot 200 + \frac{1}{3} \cdot 20^{3} + 3 \cdot 20 - (\frac{1}{3} \cdot 0^{3} + 3 \cdot 0))$

Step 12.2.3.7

Multiply $218$ by $200$ .

$A (x) = \frac{1}{20 - 0} (43600 + \frac{1}{3} \cdot 20^{3} + 3 \cdot 20 - (\frac{1}{3} \cdot 0^{3} + 3 \cdot 0))$

Step 12.2.3.8

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

$A (x) = \frac{1}{20 - 0} (43600 + \frac{1}{3} \cdot 8000 + 3 \cdot 20 - (\frac{1}{3} \cdot 0^{3} + 3 \cdot 0))$

Step 12.2.3.9

Combine $\frac{1}{3}$ and $8000$ .

$A (x) = \frac{1}{20 - 0} (43600 + \frac{8000}{3} + 3 \cdot 20 - (\frac{1}{3} \cdot 0^{3} + 3 \cdot 0))$

Step 12.2.3.10

Multiply $3$ by $20$ .

$A (x) = \frac{1}{20 - 0} (43600 + \frac{8000}{3} + 60 - (\frac{1}{3} \cdot 0^{3} + 3 \cdot 0))$

Step 12.2.3.11

To write $60$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$A (x) = \frac{1}{20 - 0} (43600 + \frac{8000}{3} + 60 \cdot \frac{3}{3} - (\frac{1}{3} \cdot 0^{3} + 3 \cdot 0))$

Step 12.2.3.12

Combine $60$ and $\frac{3}{3}$ .

$A (x) = \frac{1}{20 - 0} (43600 + \frac{8000}{3} + \frac{60 \cdot 3}{3} - (\frac{1}{3} \cdot 0^{3} + 3 \cdot 0))$

Step 12.2.3.13

Combine the numerators over the common denominator.

$A (x) = \frac{1}{20 - 0} (43600 + \frac{8000 + 60 \cdot 3}{3} - (\frac{1}{3} \cdot 0^{3} + 3 \cdot 0))$

Step 12.2.3.14

Simplify the numerator.

Tap for more steps...

Step 12.2.3.14.1

Multiply $60$ by $3$ .

$A (x) = \frac{1}{20 - 0} (43600 + \frac{8000 + 180}{3} - (\frac{1}{3} \cdot 0^{3} + 3 \cdot 0))$

Step 12.2.3.14.2

Add $8000$ and $180$ .

$A (x) = \frac{1}{20 - 0} (43600 + \frac{8180}{3} - (\frac{1}{3} \cdot 0^{3} + 3 \cdot 0))$

Step 12.2.3.15

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

$A (x) = \frac{1}{20 - 0} (43600 + \frac{8180}{3} - (\frac{1}{3} \cdot 0 + 3 \cdot 0))$

Step 12.2.3.16

Multiply $\frac{1}{3}$ by $0$ .

$A (x) = \frac{1}{20 - 0} (43600 + \frac{8180}{3} - (0 + 3 \cdot 0))$

Step 12.2.3.17

Multiply $3$ by $0$ .

$A (x) = \frac{1}{20 - 0} (43600 + \frac{8180}{3} - (0 + 0))$

Step 12.2.3.18

Add $0$ and $0$ .

$A (x) = \frac{1}{20 - 0} (43600 + \frac{8180}{3} - 0)$

Step 12.2.3.19

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{20 - 0} (43600 + \frac{8180}{3} + 0)$

Step 12.2.3.20

Add $\frac{8180}{3}$ and $0$ .

$A (x) = \frac{1}{20 - 0} (43600 + \frac{8180}{3})$

Step 12.2.3.21

To write $43600$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$A (x) = \frac{1}{20 - 0} (43600 \cdot \frac{3}{3} + \frac{8180}{3})$

Step 12.2.3.22

Combine $43600$ and $\frac{3}{3}$ .

$A (x) = \frac{1}{20 - 0} (\frac{43600 \cdot 3}{3} + \frac{8180}{3})$

Step 12.2.3.23

Combine the numerators over the common denominator.

$A (x) = \frac{1}{20 - 0} (\frac{43600 \cdot 3 + 8180}{3})$

Step 12.2.3.24

Simplify the numerator.

Tap for more steps...

Step 12.2.3.24.1

Multiply $43600$ by $3$ .

$A (x) = \frac{1}{20 - 0} (\frac{130800 + 8180}{3})$

Step 12.2.3.24.2

Add $130800$ and $8180$ .

$A (x) = \frac{1}{20 - 0} (\frac{138980}{3})$

Step 13

Simplify the denominator.

Tap for more steps...

Step 13.1

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{20 + 0} \cdot \frac{138980}{3}$

Step 13.2

Add $20$ and $0$ .

$A (x) = \frac{1}{20} \cdot \frac{138980}{3}$

Step 14

Cancel the common factor of $20$ .

Tap for more steps...

Step 14.1

Factor $20$ out of $138980$ .

$A (x) = \frac{1}{20} \cdot \frac{20 (6949)}{3}$

Step 14.2

Cancel the common factor.

$A (x) = \frac{1}{20} \cdot \frac{20 \cdot 6949}{3}$

Step 14.3

Rewrite the expression.

$A (x) = \frac{6949}{3}$

Step 15