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.

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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 $4$ .

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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.

Tap for more steps...

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 $2$ .

Tap for more steps...

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.

Tap for more steps...

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 $2$ .

Tap for more steps...

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

Enter YOUR Problem