Calculus Examples

$y = x^{2}$ , $(2, 5)$

Step 1

Write $y = x^{2}$ as a function.

$f (x) = x^{2}$

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 $[2, 5]$ .

$f (x)$ 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.

$A (x) = \frac{1}{5 - 2} (\int_{2}^{5} x^{2} d x)$

Step 6

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

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

Step 7

Substitute and simplify.

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Step 7.1

Evaluate $\frac{1}{3} x^{3}$ at $5$ and at $2$ .

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

Step 7.2

Simplify.

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Step 7.2.1

Raise $5$ to the power of $3$ .

$A (x) = \frac{1}{5 - 2} (\frac{1}{3} \cdot 125 - \frac{1}{3} \cdot 2^{3})$

Step 7.2.2

Combine $\frac{1}{3}$ and $125$ .

$A (x) = \frac{1}{5 - 2} (\frac{125}{3} - \frac{1}{3} \cdot 2^{3})$

Step 7.2.3

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

$A (x) = \frac{1}{5 - 2} (\frac{125}{3} - \frac{1}{3} \cdot 8)$

Step 7.2.4

Multiply $8$ by $- 1$ .

$A (x) = \frac{1}{5 - 2} (\frac{125}{3} - 8 (\frac{1}{3}))$

Step 7.2.5

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

$A (x) = \frac{1}{5 - 2} (\frac{125}{3} + \frac{- 8}{3})$

Step 7.2.6

Move the negative in front of the fraction.

$A (x) = \frac{1}{5 - 2} (\frac{125}{3} - \frac{8}{3})$

Step 7.2.7

Combine the numerators over the common denominator.

$A (x) = \frac{1}{5 - 2} (\frac{125 - 8}{3})$

Step 7.2.8

Subtract $8$ from $125$ .

$A (x) = \frac{1}{5 - 2} (\frac{117}{3})$

Step 7.2.9

Cancel the common factor of $117$ and $3$ .

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Step 7.2.9.1

Factor $3$ out of $117$ .

$A (x) = \frac{1}{5 - 2} (\frac{3 \cdot 39}{3})$

Step 7.2.9.2

Cancel the common factors.

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Step 7.2.9.2.1

Factor $3$ out of $3$ .

$A (x) = \frac{1}{5 - 2} (\frac{3 \cdot 39}{3 (1)})$

Step 7.2.9.2.2

Cancel the common factor.

$A (x) = \frac{1}{5 - 2} (\frac{3 \cdot 39}{3 \cdot 1})$

Step 7.2.9.2.3

Rewrite the expression.

$A (x) = \frac{1}{5 - 2} (\frac{39}{1})$

Step 7.2.9.2.4

Divide $39$ by $1$ .

$A (x) = \frac{1}{5 - 2} (39)$

Step 8

Subtract $2$ from $5$ .

$A (x) = \frac{1}{3} \cdot 39$

Step 9

Cancel the common factor of $3$ .

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Step 9.1

Factor $3$ out of $39$ .

$A (x) = \frac{1}{3} \cdot (3 (13))$

Step 9.2

Cancel the common factor.

$A (x) = \frac{1}{3} \cdot (3 \cdot 13)$

Step 9.3

Rewrite the expression.

$A (x) = 13$

Step 10