Calculus Examples

$f (x) = x^{2}$ , $[0, 2]$

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 $[0, 2]$ .

$f (x)$ 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.

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

Step 5

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

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

Step 6

Substitute and simplify.

Tap for more steps...

Step 6.1

Evaluate $\frac{1}{3} x^{3}$ at $2$ and at $0$ .

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

Step 6.2

Simplify.

Tap for more steps...

Step 6.2.1

Raise $2$ to the power of $3$ .

$A (x) = \frac{1}{2 - 0} (\frac{1}{3} \cdot 8 - \frac{1}{3} \cdot 0^{3})$

Step 6.2.2

Combine $\frac{1}{3}$ and $8$ .

$A (x) = \frac{1}{2 - 0} (\frac{8}{3} - \frac{1}{3} \cdot 0^{3})$

Step 6.2.3

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

$A (x) = \frac{1}{2 - 0} (\frac{8}{3} - \frac{1}{3} \cdot 0)$

Step 6.2.4

Multiply $0$ by $- 1$ .

$A (x) = \frac{1}{2 - 0} (\frac{8}{3} + 0 (\frac{1}{3}))$

Step 6.2.5

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

$A (x) = \frac{1}{2 - 0} (\frac{8}{3} + 0)$

Step 6.2.6

Add $\frac{8}{3}$ and $0$ .

$A (x) = \frac{1}{2 - 0} (\frac{8}{3})$

Step 7

Simplify the denominator.

Tap for more steps...

Step 7.1

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{2 + 0} \cdot \frac{8}{3}$

Step 7.2

Add $2$ and $0$ .

$A (x) = \frac{1}{2} \cdot \frac{8}{3}$

Step 8

Cancel the common factor of $2$ .

Tap for more steps...

Step 8.1

Factor $2$ out of $8$ .

$A (x) = \frac{1}{2} \cdot \frac{2 (4)}{3}$

Step 8.2

Cancel the common factor.

$A (x) = \frac{1}{2} \cdot \frac{2 \cdot 4}{3}$

Step 8.3

Rewrite the expression.

$A (x) = \frac{4}{3}$

Step 9