Calculus Examples

$f (x) = 5 x - 3$ ,

$[0, 6]$

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

$[0, 6]$ .

$f (x)$ 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.

$A (x) = \frac{1}{6 - 0} (\int_{0}^{6} 5 x - 3 d x)$

Step 5

Split the single integral into multiple integrals.

$A (x) = \frac{1}{6 - 0} (\int_{0}^{6} 5 x d x + \int_{0}^{6} - 3 d x)$

Step 6

Since

$5$ is constant with respect to

$x$ , move

$5$ out of the integral.

$A (x) = \frac{1}{6 - 0} (5 \int_{0}^{6} x d x + \int_{0}^{6} - 3 d x)$

Step 7

By the Power Rule, the integral of

$x$ with respect to

$x$ is

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

$A (x) = \frac{1}{6 - 0} (5 (\frac{1}{2} x^{2}]_{0}^{6}) + \int_{0}^{6} - 3 d x)$

Step 8

Combine

$\frac{1}{2}$ and

$x^{2}$ .

$A (x) = \frac{1}{6 - 0} (5 (\frac{x^{2}}{2}]_{0}^{6}) + \int_{0}^{6} - 3 d x)$

Step 9

Apply the constant rule.

$A (x) = \frac{1}{6 - 0} (5 (\frac{x^{2}}{2}]_{0}^{6}) + - 3 x]_{0}^{6})$

Step 10

Substitute and simplify.

Tap for more steps...

Step 10.1

Evaluate

$\frac{x^{2}}{2}$ at

$6$ and at

$0$ .

$A (x) = \frac{1}{6 - 0} (5 ((\frac{6^{2}}{2}) - \frac{0^{2}}{2}) + - 3 x]_{0}^{6})$

Step 10.2

Evaluate

$- 3 x$ at

$6$ and at

$0$ .

$A (x) = \frac{1}{6 - 0} (5 (\frac{6^{2}}{2} - \frac{0^{2}}{2}) - 3 \cdot 6 + 3 \cdot 0)$

Step 10.3

Simplify.

Tap for more steps...

Step 10.3.1

Raise

$6$ to the power of

$2$ .

$A (x) = \frac{1}{6 - 0} (5 (\frac{36}{2} - \frac{0^{2}}{2}) - 3 \cdot 6 + 3 \cdot 0)$

Step 10.3.2

Cancel the common factor of

$36$ and

$2$ .

Tap for more steps...

Step 10.3.2.1

Factor

$2$ out of

$36$ .

$A (x) = \frac{1}{6 - 0} (5 (\frac{2 \cdot 18}{2} - \frac{0^{2}}{2}) - 3 \cdot 6 + 3 \cdot 0)$

Step 10.3.2.2

Cancel the common factors.

Tap for more steps...

Step 10.3.2.2.1

Factor

$2$ out of

$2$ .

$A (x) = \frac{1}{6 - 0} (5 (\frac{2 \cdot 18}{2 (1)} - \frac{0^{2}}{2}) - 3 \cdot 6 + 3 \cdot 0)$

Step 10.3.2.2.2

Cancel the common factor.

$A (x) = \frac{1}{6 - 0} (5 (\frac{2 \cdot 18}{2 \cdot 1} - \frac{0^{2}}{2}) - 3 \cdot 6 + 3 \cdot 0)$

Step 10.3.2.2.3

Rewrite the expression.

$A (x) = \frac{1}{6 - 0} (5 (\frac{18}{1} - \frac{0^{2}}{2}) - 3 \cdot 6 + 3 \cdot 0)$

Step 10.3.2.2.4

Divide

$18$ by

$1$ .

$A (x) = \frac{1}{6 - 0} (5 (18 - \frac{0^{2}}{2}) - 3 \cdot 6 + 3 \cdot 0)$

Step 10.3.3

Raising

$0$ to any positive power yields

$0$ .

$A (x) = \frac{1}{6 - 0} (5 (18 - \frac{0}{2}) - 3 \cdot 6 + 3 \cdot 0)$

Step 10.3.4

Cancel the common factor of

$0$ and

$2$ .

Tap for more steps...

Step 10.3.4.1

Factor

$2$ out of

$0$ .

$A (x) = \frac{1}{6 - 0} (5 (18 - \frac{2 (0)}{2}) - 3 \cdot 6 + 3 \cdot 0)$

Step 10.3.4.2

Cancel the common factors.

Tap for more steps...

Step 10.3.4.2.1

Factor

$2$ out of

$2$ .

$A (x) = \frac{1}{6 - 0} (5 (18 - \frac{2 \cdot 0}{2 \cdot 1}) - 3 \cdot 6 + 3 \cdot 0)$

Step 10.3.4.2.2

Cancel the common factor.

$A (x) = \frac{1}{6 - 0} (5 (18 - \frac{2 \cdot 0}{2 \cdot 1}) - 3 \cdot 6 + 3 \cdot 0)$

Step 10.3.4.2.3

Rewrite the expression.

$A (x) = \frac{1}{6 - 0} (5 (18 - \frac{0}{1}) - 3 \cdot 6 + 3 \cdot 0)$

Step 10.3.4.2.4

Divide

$0$ by

$1$ .

$A (x) = \frac{1}{6 - 0} (5 (18 - 0) - 3 \cdot 6 + 3 \cdot 0)$

Step 10.3.5

Multiply

$- 1$ by

$0$ .

$A (x) = \frac{1}{6 - 0} (5 (18 + 0) - 3 \cdot 6 + 3 \cdot 0)$

Step 10.3.6

Add

$18$ and

$0$ .

$A (x) = \frac{1}{6 - 0} (5 \cdot 18 - 3 \cdot 6 + 3 \cdot 0)$

Step 10.3.7

Multiply

$5$ by

$18$ .

$A (x) = \frac{1}{6 - 0} (90 - 3 \cdot 6 + 3 \cdot 0)$

Step 10.3.8

Multiply

$- 3$ by

$6$ .

$A (x) = \frac{1}{6 - 0} (90 - 18 + 3 \cdot 0)$

Step 10.3.9

Multiply

$3$ by

$0$ .

$A (x) = \frac{1}{6 - 0} (90 - 18 + 0)$

Step 10.3.10

Add

$- 18$ and

$0$ .

$A (x) = \frac{1}{6 - 0} (90 - 18)$

Step 10.3.11

Subtract

$18$ from

$90$ .

$A (x) = \frac{1}{6 - 0} (72)$

Step 11

Simplify the denominator.

Tap for more steps...

Step 11.1

Multiply

$- 1$ by

$0$ .

$A (x) = \frac{1}{6 + 0} \cdot 72$

Step 11.2

Add

$6$ and

$0$ .

$A (x) = \frac{1}{6} \cdot 72$

Step 12

Cancel the common factor of

$6$ .

Tap for more steps...

Step 12.1

Factor

$6$ out of

$72$ .

$A (x) = \frac{1}{6} \cdot (6 (12))$

Step 12.2

Cancel the common factor.

$A (x) = \frac{1}{6} \cdot (6 \cdot 12)$

Step 12.3

Rewrite the expression.

$A (x) = 12$

Step 13

Enter YOUR Problem