Calculus Examples

$y = 3 x^{3} + x + 3$ ,

$(5, 7)$

Step 1

Write

$y = 3 x^{3} + x + 3$ as a function.

$f (x) = 3 x^{3} + x + 3$

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

$[5, 7]$ .

$f (x)$ 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.

$A (x) = \frac{1}{7 - 5} (\int_{5}^{7} 3 x^{3} + x + 3 d x)$

Step 6

Split the single integral into multiple integrals.

$A (x) = \frac{1}{7 - 5} (\int_{5}^{7} 3 x^{3} d x + \int_{5}^{7} x d x + \int_{5}^{7} 3 d x)$

Step 7

Since

$3$ is constant with respect to

$x$ , move

$3$ out of the integral.

$A (x) = \frac{1}{7 - 5} (3 \int_{5}^{7} x^{3} d x + \int_{5}^{7} x d x + \int_{5}^{7} 3 d x)$

Step 8

By the Power Rule, the integral of

$x^{3}$ with respect to

$x$ is

$\frac{1}{4} x^{4}$ .

$A (x) = \frac{1}{7 - 5} (3 (\frac{1}{4} x^{4}]_{5}^{7}) + \int_{5}^{7} x d x + \int_{5}^{7} 3 d x)$

Step 9

Combine

$\frac{1}{4}$ and

$x^{4}$ .

$A (x) = \frac{1}{7 - 5} (3 (\frac{x^{4}}{4}]_{5}^{7}) + \int_{5}^{7} x d x + \int_{5}^{7} 3 d x)$

Step 10

By the Power Rule, the integral of

$x$ with respect to

$x$ is

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

$A (x) = \frac{1}{7 - 5} (3 (\frac{x^{4}}{4}]_{5}^{7}) + \frac{1}{2} x^{2}]_{5}^{7} + \int_{5}^{7} 3 d x)$

Step 11

Apply the constant rule.

$A (x) = \frac{1}{7 - 5} (3 (\frac{x^{4}}{4}]_{5}^{7}) + \frac{1}{2} x^{2}]_{5}^{7} + 3 x]_{5}^{7})$

Step 12

Simplify the answer.

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Step 12.1

Combine

$\frac{1}{2} x^{2}]_{5}^{7}$ and

$3 x]_{5}^{7}$ .

$A (x) = \frac{1}{7 - 5} (3 (\frac{x^{4}}{4}]_{5}^{7}) + \frac{1}{2} x^{2} + 3 x]_{5}^{7})$

Step 12.2

Substitute and simplify.

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Step 12.2.1

Evaluate

$\frac{x^{4}}{4}$ at

$7$ and at

$5$ .

$A (x) = \frac{1}{7 - 5} (3 ((\frac{7^{4}}{4}) - \frac{5^{4}}{4}) + \frac{1}{2} x^{2} + 3 x]_{5}^{7})$

Step 12.2.2

Evaluate

$\frac{1}{2} x^{2} + 3 x$ at

$7$ and at

$5$ .

$A (x) = \frac{1}{7 - 5} (3 ((\frac{7^{4}}{4}) - \frac{5^{4}}{4}) + \frac{1}{2} \cdot 7^{2} + 3 \cdot 7 - (\frac{1}{2} \cdot 5^{2} + 3 \cdot 5))$

Step 12.2.3

Simplify.

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Step 12.2.3.1

Raise

$7$ to the power of

$4$ .

$A (x) = \frac{1}{7 - 5} (3 (\frac{2401}{4} - \frac{5^{4}}{4}) + \frac{1}{2} \cdot 7^{2} + 3 \cdot 7 - (\frac{1}{2} \cdot 5^{2} + 3 \cdot 5))$

Step 12.2.3.2

Raise

$5$ to the power of

$4$ .

$A (x) = \frac{1}{7 - 5} (3 (\frac{2401}{4} - \frac{625}{4}) + \frac{1}{2} \cdot 7^{2} + 3 \cdot 7 - (\frac{1}{2} \cdot 5^{2} + 3 \cdot 5))$

Step 12.2.3.3

Combine the numerators over the common denominator.

$A (x) = \frac{1}{7 - 5} (3 (\frac{2401 - 625}{4}) + \frac{1}{2} \cdot 7^{2} + 3 \cdot 7 - (\frac{1}{2} \cdot 5^{2} + 3 \cdot 5))$

Step 12.2.3.4

Subtract

$625$ from

$2401$ .

$A (x) = \frac{1}{7 - 5} (3 (\frac{1776}{4}) + \frac{1}{2} \cdot 7^{2} + 3 \cdot 7 - (\frac{1}{2} \cdot 5^{2} + 3 \cdot 5))$

Step 12.2.3.5

Cancel the common factor of

$1776$ and

$4$ .

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Step 12.2.3.5.1

Factor

$4$ out of

$1776$ .

$A (x) = \frac{1}{7 - 5} (3 (\frac{4 \cdot 444}{4}) + \frac{1}{2} \cdot 7^{2} + 3 \cdot 7 - (\frac{1}{2} \cdot 5^{2} + 3 \cdot 5))$

Step 12.2.3.5.2

Cancel the common factors.

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Step 12.2.3.5.2.1

Factor

$4$ out of

$4$ .

$A (x) = \frac{1}{7 - 5} (3 (\frac{4 \cdot 444}{4 (1)}) + \frac{1}{2} \cdot 7^{2} + 3 \cdot 7 - (\frac{1}{2} \cdot 5^{2} + 3 \cdot 5))$

Step 12.2.3.5.2.2

Cancel the common factor.

$A (x) = \frac{1}{7 - 5} (3 (\frac{4 \cdot 444}{4 \cdot 1}) + \frac{1}{2} \cdot 7^{2} + 3 \cdot 7 - (\frac{1}{2} \cdot 5^{2} + 3 \cdot 5))$

Step 12.2.3.5.2.3

Rewrite the expression.

$A (x) = \frac{1}{7 - 5} (3 (\frac{444}{1}) + \frac{1}{2} \cdot 7^{2} + 3 \cdot 7 - (\frac{1}{2} \cdot 5^{2} + 3 \cdot 5))$

Step 12.2.3.5.2.4

Divide

$444$ by

$1$ .

$A (x) = \frac{1}{7 - 5} (3 \cdot 444 + \frac{1}{2} \cdot 7^{2} + 3 \cdot 7 - (\frac{1}{2} \cdot 5^{2} + 3 \cdot 5))$

Step 12.2.3.6

Multiply

$3$ by

$444$ .

$A (x) = \frac{1}{7 - 5} (1332 + \frac{1}{2} \cdot 7^{2} + 3 \cdot 7 - (\frac{1}{2} \cdot 5^{2} + 3 \cdot 5))$

Step 12.2.3.7

Raise

$7$ to the power of

$2$ .

$A (x) = \frac{1}{7 - 5} (1332 + \frac{1}{2} \cdot 49 + 3 \cdot 7 - (\frac{1}{2} \cdot 5^{2} + 3 \cdot 5))$

Step 12.2.3.8

Combine

$\frac{1}{2}$ and

$49$ .

$A (x) = \frac{1}{7 - 5} (1332 + \frac{49}{2} + 3 \cdot 7 - (\frac{1}{2} \cdot 5^{2} + 3 \cdot 5))$

Step 12.2.3.9

Multiply

$3$ by

$7$ .

$A (x) = \frac{1}{7 - 5} (1332 + \frac{49}{2} + 21 - (\frac{1}{2} \cdot 5^{2} + 3 \cdot 5))$

Step 12.2.3.10

To write

$21$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$A (x) = \frac{1}{7 - 5} (1332 + \frac{49}{2} + 21 \cdot \frac{2}{2} - (\frac{1}{2} \cdot 5^{2} + 3 \cdot 5))$

Step 12.2.3.11

Combine

$21$ and

$\frac{2}{2}$ .

$A (x) = \frac{1}{7 - 5} (1332 + \frac{49}{2} + \frac{21 \cdot 2}{2} - (\frac{1}{2} \cdot 5^{2} + 3 \cdot 5))$

Step 12.2.3.12

Combine the numerators over the common denominator.

$A (x) = \frac{1}{7 - 5} (1332 + \frac{49 + 21 \cdot 2}{2} - (\frac{1}{2} \cdot 5^{2} + 3 \cdot 5))$

Step 12.2.3.13

Simplify the numerator.

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Step 12.2.3.13.1

Multiply

$21$ by

$2$ .

$A (x) = \frac{1}{7 - 5} (1332 + \frac{49 + 42}{2} - (\frac{1}{2} \cdot 5^{2} + 3 \cdot 5))$

Step 12.2.3.13.2

Add

$49$ and

$42$ .

$A (x) = \frac{1}{7 - 5} (1332 + \frac{91}{2} - (\frac{1}{2} \cdot 5^{2} + 3 \cdot 5))$

Step 12.2.3.14

Raise

$5$ to the power of

$2$ .

$A (x) = \frac{1}{7 - 5} (1332 + \frac{91}{2} - (\frac{1}{2} \cdot 25 + 3 \cdot 5))$

Step 12.2.3.15

Combine

$\frac{1}{2}$ and

$25$ .

$A (x) = \frac{1}{7 - 5} (1332 + \frac{91}{2} - (\frac{25}{2} + 3 \cdot 5))$

Step 12.2.3.16

Multiply

$3$ by

$5$ .

$A (x) = \frac{1}{7 - 5} (1332 + \frac{91}{2} - (\frac{25}{2} + 15))$

Step 12.2.3.17

To write

$15$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$A (x) = \frac{1}{7 - 5} (1332 + \frac{91}{2} - (\frac{25}{2} + 15 \cdot \frac{2}{2}))$

Step 12.2.3.18

Combine

$15$ and

$\frac{2}{2}$ .

$A (x) = \frac{1}{7 - 5} (1332 + \frac{91}{2} - (\frac{25}{2} + \frac{15 \cdot 2}{2}))$

Step 12.2.3.19

Combine the numerators over the common denominator.

$A (x) = \frac{1}{7 - 5} (1332 + \frac{91}{2} - \frac{25 + 15 \cdot 2}{2})$

Step 12.2.3.20

Simplify the numerator.

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Step 12.2.3.20.1

Multiply

$15$ by

$2$ .

$A (x) = \frac{1}{7 - 5} (1332 + \frac{91}{2} - \frac{25 + 30}{2})$

Step 12.2.3.20.2

Add

$25$ and

$30$ .

$A (x) = \frac{1}{7 - 5} (1332 + \frac{91}{2} - \frac{55}{2})$

Step 12.2.3.21

Combine the numerators over the common denominator.

$A (x) = \frac{1}{7 - 5} (1332 + \frac{91 - 55}{2})$

Step 12.2.3.22

Subtract

$55$ from

$91$ .

$A (x) = \frac{1}{7 - 5} (1332 + \frac{36}{2})$

Step 12.2.3.23

Cancel the common factor of

$36$ and

$2$ .

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Step 12.2.3.23.1

Factor

$2$ out of

$36$ .

$A (x) = \frac{1}{7 - 5} (1332 + \frac{2 \cdot 18}{2})$

Step 12.2.3.23.2

Cancel the common factors.

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Step 12.2.3.23.2.1

Factor

$2$ out of

$2$ .

$A (x) = \frac{1}{7 - 5} (1332 + \frac{2 \cdot 18}{2 (1)})$

Step 12.2.3.23.2.2

Cancel the common factor.

$A (x) = \frac{1}{7 - 5} (1332 + \frac{2 \cdot 18}{2 \cdot 1})$

Step 12.2.3.23.2.3

Rewrite the expression.

$A (x) = \frac{1}{7 - 5} (1332 + \frac{18}{1})$

Step 12.2.3.23.2.4

Divide

$18$ by

$1$ .

$A (x) = \frac{1}{7 - 5} (1332 + 18)$

Step 12.2.3.24

Add

$1332$ and

$18$ .

$A (x) = \frac{1}{7 - 5} (1350)$

Step 13

Subtract

$5$ from

$7$ .

$A (x) = \frac{1}{2} \cdot 1350$

Step 14

Cancel the common factor of

$2$ .

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Step 14.1

Factor

$2$ out of

$1350$ .

$A (x) = \frac{1}{2} \cdot (2 (675))$

Step 14.2

Cancel the common factor.

$A (x) = \frac{1}{2} \cdot (2 \cdot 675)$

Step 14.3

Rewrite the expression.

$A (x) = 675$

Step 15

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