Calculus Examples

$f (x) = x^{3}$ , $[0, 1]$

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 $[0, 1]$ .

$f (x)$ 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.

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

Step 5

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$A (x) = \frac{1}{1 - 0} (\frac{1}{4} x^{4}]_{0}^{1})$

Step 6

Substitute and simplify.

Tap for more steps...

Step 6.1

Evaluate $\frac{1}{4} x^{4}$ at $1$ and at $0$ .

$A (x) = \frac{1}{1 - 0} ((\frac{1}{4} \cdot 1^{4}) - \frac{1}{4} \cdot 0^{4})$

Step 6.2

Simplify.

Tap for more steps...

Step 6.2.1

One to any power is one.

$A (x) = \frac{1}{1 - 0} (\frac{1}{4} \cdot 1 - \frac{1}{4} \cdot 0^{4})$

Step 6.2.2

Multiply $\frac{1}{4}$ by $1$ .

$A (x) = \frac{1}{1 - 0} (\frac{1}{4} - \frac{1}{4} \cdot 0^{4})$

Step 6.2.3

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

$A (x) = \frac{1}{1 - 0} (\frac{1}{4} - \frac{1}{4} \cdot 0)$

Step 6.2.4

Multiply $0$ by $- 1$ .

$A (x) = \frac{1}{1 - 0} (\frac{1}{4} + 0 (\frac{1}{4}))$

Step 6.2.5

Multiply $0$ by $\frac{1}{4}$ .

$A (x) = \frac{1}{1 - 0} (\frac{1}{4} + 0)$

Step 6.2.6

Add $\frac{1}{4}$ and $0$ .

$A (x) = \frac{1}{1 - 0} (\frac{1}{4})$

Step 7

Simplify the denominator.

Tap for more steps...

Step 7.1

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{1 + 0} \cdot \frac{1}{4}$

Step 7.2

Add $1$ and $0$ .

$A (x) = \frac{1}{1} \cdot \frac{1}{4}$

Step 8

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 8.1

Cancel the common factor of $1$ .

Tap for more steps...

Step 8.1.1

Cancel the common factor.

$A (x) = \frac{1}{1} \cdot \frac{1}{4}$

Step 8.1.2

Rewrite the expression.

$A (x) = 1 \cdot \frac{1}{4}$

Step 8.2

Multiply $\frac{1}{4}$ by $1$ .

$A (x) = \frac{1}{4}$

Step 9