Calculus Examples

$f (x) = x \cdot e^{x}$ , $[0, 1]$

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.

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

Step 5

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x$ and $d v = e^{x}$ .

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

Step 6

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

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

Step 7

Substitute and simplify.

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Step 7.1

Evaluate $x e^{x}$ at $1$ and at $0$ .

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

Step 7.2

Evaluate $e^{x}$ at $1$ and at $0$ .

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

Step 7.3

Simplify.

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Step 7.3.1

Simplify.

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

Step 7.3.2

Multiply $e$ by $1$ .

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

Step 7.3.3

Anything raised to $0$ is $1$ .

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

Step 7.3.4

Multiply $0$ by $1$ .

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

Step 7.3.5

Add $e$ and $0$ .

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

Step 7.3.6

Simplify.

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

Step 7.3.7

Anything raised to $0$ is $1$ .

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

Step 7.3.8

Multiply $- 1$ by $1$ .

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

Step 8

Simplify.

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Step 8.1

Simplify each term.

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Step 8.1.1

Apply the distributive property.

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

Step 8.1.2

Multiply $- 1$ by $- 1$ .

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

Step 8.2

Subtract $e$ from $e$ .

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

Step 8.3

Add $0$ and $1$ .

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

Step 9

Simplify the denominator.

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Step 9.1

Multiply $- 1$ by $0$ .

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

Step 9.2

Add $1$ and $0$ .

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

Step 10

Reduce the expression by cancelling the common factors.

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Step 10.1

Cancel the common factor of $1$ .

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Step 10.1.1

Cancel the common factor.

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

Step 10.1.2

Rewrite the expression.

$A (x) = 1 \cdot 1$

Step 10.2

Multiply $1$ by $1$ .

$A (x) = 1$

Step 11