Calculus Examples

$y = e^{2 x}$ ; $[0, 4]$

Step 1

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

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

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 $[0, 4]$ .

$f (x)$ 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.

$A (x) = \frac{1}{4 - 0} (\int_{0}^{4} e^{2 x} d x)$

Step 6

Let $u = 2 x$ . Then $d u = 2 d x$ , so $\frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 6.1

Let $u = 2 x$ . Find $\frac{d u}{d x}$ .

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Step 6.1.1

Differentiate $2 x$ .

$\frac{d}{d x} [2 x]$

Step 6.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x]$

Step 6.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 \cdot 1$

Step 6.1.4

Multiply $2$ by $1$ .

$2$

Step 6.2

Substitute the lower limit in for $x$ in $u = 2 x$ .

$u_{lower} = 2 \cdot 0$

Step 6.3

Multiply $2$ by $0$ .

$u_{lower} = 0$

Step 6.4

Substitute the upper limit in for $x$ in $u = 2 x$ .

$u_{upper} = 2 \cdot 4$

Step 6.5

Multiply $2$ by $4$ .

$u_{upper} = 8$

Step 6.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 0$

$u_{upper} = 8$

Step 6.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$A (x) = \frac{1}{4 - 0} (\int_{0}^{8} e^{u} (\frac{1}{2}) d u)$

Step 7

Combine $e^{u}$ and $\frac{1}{2}$ .

$A (x) = \frac{1}{4 - 0} (\int_{0}^{8} \frac{e^{u}}{2} d u)$

Step 8

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$A (x) = \frac{1}{4 - 0} (\frac{1}{2} \cdot \int_{0}^{8} e^{u} d u)$

Step 9

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$A (x) = \frac{1}{4 - 0} (\frac{1}{2} \cdot e^{u}]_{0}^{8})$

Step 10

Substitute and simplify.

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Step 10.1

Evaluate $e^{u}$ at $8$ and at $0$ .

$A (x) = \frac{1}{4 - 0} (\frac{1}{2} \cdot ((e^{8}) - e^{0}))$

Step 10.2

Simplify.

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Step 10.2.1

Anything raised to $0$ is $1$ .

$A (x) = \frac{1}{4 - 0} (\frac{1}{2} \cdot (e^{8} - 1 \cdot 1))$

Step 10.2.2

Multiply $- 1$ by $1$ .

$A (x) = \frac{1}{4 - 0} (\frac{1}{2} \cdot (e^{8} - 1))$

Step 11

Simplify.

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Step 11.1

Apply the distributive property.

$A (x) = \frac{1}{4 - 0} (\frac{1}{2} \cdot e^{8} + \frac{1}{2} \cdot - 1)$

Step 11.2

Combine $\frac{1}{2}$ and $e^{8}$ .

$A (x) = \frac{1}{4 - 0} (\frac{e^{8}}{2} + \frac{1}{2} \cdot - 1)$

Step 11.3

Combine $\frac{1}{2}$ and $- 1$ .

$A (x) = \frac{1}{4 - 0} (\frac{e^{8}}{2} + \frac{- 1}{2})$

Step 11.4

Move the negative in front of the fraction.

$A (x) = \frac{1}{4 - 0} (\frac{e^{8}}{2} - \frac{1}{2})$

Step 12

Simplify the denominator.

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Step 12.1

Multiply $- 1$ by $0$ .

$A (x) = \frac{1}{4 + 0} \cdot (\frac{e^{8}}{2} - \frac{1}{2})$

Step 12.2

Add $4$ and $0$ .

$A (x) = \frac{1}{4} \cdot (\frac{e^{8}}{2} - \frac{1}{2})$

Step 13

Simplify terms.

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Step 13.1

Apply the distributive property.

$A (x) = \frac{1}{4} \cdot \frac{e^{8}}{2} + \frac{1}{4} \cdot (- \frac{1}{2})$

Step 13.2

Combine.

$A (x) = \frac{1 e^{8}}{4 \cdot 2} + \frac{1}{4} \cdot (- \frac{1}{2})$

Step 14

Multiply $\frac{1}{4} (- \frac{1}{2})$ .

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Step 14.1

Multiply $\frac{1}{4}$ by $\frac{1}{2}$ .

$A (x) = \frac{1 e^{8}}{4 \cdot 2} - \frac{1}{4 \cdot 2}$

Step 14.2

Multiply $4$ by $2$ .

$A (x) = \frac{1 e^{8}}{4 \cdot 2} - \frac{1}{8}$

Step 15

Simplify each term.

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Step 15.1

Multiply $e^{8}$ by $1$ .

$A (x) = \frac{e^{8}}{4 \cdot 2} - \frac{1}{8}$

Step 15.2

Multiply $4$ by $2$ .

$A (x) = \frac{e^{8}}{8} - \frac{1}{8}$

Step 16