Calculus Examples

$y = \frac{1}{2} (e^{x} + e^{- x})$ , $[0, 2]$

Step 1

Solve by substitution to find the intersection between the curves.

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Step 1.1

Eliminate the equal sides of each equation and combine.

$\frac{e^{x}}{2} + \frac{e^{- x}}{2} = 0$

Step 1.2

Solve $\frac{e^{x}}{2} + \frac{e^{- x}}{2} = 0$ for $x$ .

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Step 1.2.1

Move $\frac{e^{- x}}{2}$ to the right side of the equation by subtracting it from both sides.

$\frac{e^{x}}{2} = - \frac{e^{- x}}{2}$

Step 1.2.2

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$e^{x} = - e^{- x}$

Step 1.2.3

Graph each side of the equation. The solution is the x-value of the point of intersection.

No solution

Step 2

Simplify $\frac{1}{2} \cdot (e^{x} + e^{- x})$ .

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Step 2.1

Apply the distributive property.

$y = \frac{1}{2} e^{x} + \frac{1}{2} e^{- x}$

Step 2.2

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

$y = \frac{e^{x}}{2} + \frac{1}{2} e^{- x}$

Step 2.3

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

$y = \frac{e^{x}}{2} + \frac{e^{- x}}{2}$

Step 3

The area of the region between the curves is defined as the integral of the upper curve minus the integral of the lower curve over each region. The regions are determined by the intersection points of the curves. This can be done algebraically or graphically.

$A r e a = \int_{0}^{2} \frac{e^{x}}{2} + \frac{e^{- x}}{2} d x - \int_{0}^{2} 0 d x$

Step 4

Integrate to find the area between $0$ and $2$ .

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Step 4.1

Combine the integrals into a single integral.

$\int_{0}^{2} \frac{e^{x}}{2} + \frac{e^{- x}}{2} - (0) d x$

Step 4.2

Subtract $0$ from $\frac{e^{x}}{2} + \frac{e^{- x}}{2}$ .

$\int_{0}^{2} \frac{e^{x}}{2} + \frac{e^{- x}}{2} d x$

Step 4.3

Split the single integral into multiple integrals.

$\int_{0}^{2} \frac{e^{x}}{2} d x + \int_{0}^{2} \frac{e^{- x}}{2} d x$

Step 4.4

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

$\frac{1}{2} \int_{0}^{2} e^{x} d x + \int_{0}^{2} \frac{e^{- x}}{2} d x$

Step 4.5

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

$\frac{1}{2} e^{x}]_{0}^{2} + \int_{0}^{2} \frac{e^{- x}}{2} d x$

Step 4.6

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

$\frac{1}{2} e^{x}]_{0}^{2} + \frac{1}{2} \int_{0}^{2} e^{- x} d x$

Step 4.7

Let $u = - x$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.7.1

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

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Step 4.7.1.1

Differentiate $- x$ .

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

Step 4.7.1.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

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

Step 4.7.1.3

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

$- 1 \cdot 1$

Step 4.7.1.4

Multiply $- 1$ by $1$ .

$- 1$

Step 4.7.2

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

$u_{lower} = - 0$

Step 4.7.3

Multiply $- 1$ by $0$ .

$u_{lower} = 0$

Step 4.7.4

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

$u_{upper} = - 1 \cdot 2$

Step 4.7.5

Multiply $- 1$ by $2$ .

$u_{upper} = - 2$

Step 4.7.6

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

$u_{lower} = 0$

$u_{upper} = - 2$

Step 4.7.7

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

$\frac{1}{2} e^{x}]_{0}^{2} + \frac{1}{2} \int_{0}^{- 2} - e^{u} d u$

Step 4.8

Since $- 1$ is constant with respect to $u$ , move $- 1$ out of the integral.

$\frac{1}{2} e^{x}]_{0}^{2} + \frac{1}{2} (- \int_{0}^{- 2} e^{u} d u)$

Step 4.9

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

$\frac{1}{2} e^{x}]_{0}^{2} + \frac{1}{2} (- (e^{u}]_{0}^{- 2}))$

Step 4.10

Combine $\frac{1}{2}$ and $e^{u}]_{0}^{- 2}$ .

$\frac{1}{2} e^{x}]_{0}^{2} - \frac{e^{u}]_{0}^{- 2}}{2}$

Step 4.11

Substitute and simplify.

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Step 4.11.1

Evaluate $e^{x}$ at $2$ and at $0$ .

$\frac{1}{2} ((e^{2}) - e^{0}) - \frac{e^{u}]_{0}^{- 2}}{2}$

Step 4.11.2

Evaluate $e^{u}$ at $- 2$ and at $0$ .

$\frac{1}{2} ((e^{2}) - e^{0}) - \frac{(e^{- 2}) - e^{0}}{2}$

Step 4.11.3

Simplify.

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Step 4.11.3.1

Anything raised to $0$ is $1$ .

$\frac{1}{2} (e^{2} - 1 \cdot 1) - \frac{(e^{- 2}) - e^{0}}{2}$

Step 4.11.3.2

Multiply $- 1$ by $1$ .

$\frac{1}{2} (e^{2} - 1) - \frac{(e^{- 2}) - e^{0}}{2}$

Step 4.11.3.3

Anything raised to $0$ is $1$ .

$\frac{1}{2} (e^{2} - 1) - \frac{e^{- 2} - 1 \cdot 1}{2}$

Step 4.11.3.4

Multiply $- 1$ by $1$ .

$\frac{1}{2} (e^{2} - 1) - \frac{e^{- 2} - 1}{2}$

Step 4.11.3.5

To write $\frac{1}{2} (e^{2} - 1)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} (e^{2} - 1) \cdot \frac{2}{2} - \frac{e^{- 2} - 1}{2}$

Step 4.11.3.6

Combine $\frac{1}{2} (e^{2} - 1)$ and $\frac{2}{2}$ .

$\frac{\frac{1}{2} (e^{2} - 1) \cdot 2}{2} - \frac{e^{- 2} - 1}{2}$

Step 4.11.3.7

Combine the numerators over the common denominator.

$\frac{\frac{1}{2} (e^{2} - 1) \cdot 2 - (e^{- 2} - 1)}{2}$

Step 4.11.3.8

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

$\frac{\frac{2}{2} (e^{2} - 1) - (e^{- 2} - 1)}{2}$

Step 4.11.3.9

Cancel the common factor of $2$ .

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Step 4.11.3.9.1

Cancel the common factor.

$\frac{\frac{2}{2} (e^{2} - 1) - (e^{- 2} - 1)}{2}$

Step 4.11.3.9.2

Rewrite the expression.

$\frac{1 (e^{2} - 1) - (e^{- 2} - 1)}{2}$

Step 4.11.3.10

Multiply $e^{2} - 1$ by $1$ .

$\frac{e^{2} - 1 - (e^{- 2} - 1)}{2}$

Step 5

Add the areas $\frac{e^{2} - 1 - (e^{- 2} - 1)}{2}$ .

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{e^{2} - 1 - (\frac{1}{e^{2}} - 1)}{2}$

Step 5.1.2

Apply the distributive property.

$\frac{e^{2} - 1 - \frac{1}{e^{2}} - - 1}{2}$

Step 5.1.3

Multiply $- 1$ by $- 1$ .

$\frac{e^{2} - 1 - \frac{1}{e^{2}} + 1}{2}$

Step 5.1.4

Add $- 1$ and $1$ .

$\frac{e^{2} - \frac{1}{e^{2}} + 0}{2}$

Step 5.1.5

Add $e^{2} - \frac{1}{e^{2}}$ and $0$ .

$\frac{e^{2} - \frac{1}{e^{2}}}{2}$

Step 5.1.6

Rewrite $e^{2} - \frac{1}{e^{2}}$ in a factored form.

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Step 5.1.6.1

Rewrite $\frac{1}{e^{2}}$ as ${(\frac{1}{e})}^{2}$ .

$\frac{e^{2} - {(\frac{1}{e})}^{2}}{2}$

Step 5.1.6.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = e$ and $b = \frac{1}{e}$ .

$\frac{(e + \frac{1}{e}) (e - \frac{1}{e})}{2}$

Step 5.1.7

To write $e$ as a fraction with a common denominator, multiply by $\frac{e}{e}$ .

$\frac{(\frac{e e}{e} + \frac{1}{e}) (e - \frac{1}{e})}{2}$

Step 5.1.8

Combine the numerators over the common denominator.

$\frac{\frac{e e + 1}{e} (e - \frac{1}{e})}{2}$

Step 5.1.9

Multiply $e e$ .

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Step 5.1.9.1

Raise $e$ to the power of $1$ .

$\frac{\frac{e^{1} e + 1}{e} (e - \frac{1}{e})}{2}$

Step 5.1.9.2

Raise $e$ to the power of $1$ .

$\frac{\frac{e^{1} e^{1} + 1}{e} (e - \frac{1}{e})}{2}$

Step 5.1.9.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{e^{1 + 1} + 1}{e} (e - \frac{1}{e})}{2}$

Step 5.1.9.4

Add $1$ and $1$ .

$\frac{\frac{e^{2} + 1}{e} (e - \frac{1}{e})}{2}$

Step 5.1.10

To write $e$ as a fraction with a common denominator, multiply by $\frac{e}{e}$ .

$\frac{\frac{e^{2} + 1}{e} (\frac{e e}{e} - \frac{1}{e})}{2}$

Step 5.1.11

Combine the numerators over the common denominator.

$\frac{\frac{e^{2} + 1}{e} \cdot \frac{e e - 1}{e}}{2}$

Step 5.1.12

Simplify the numerator.

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Step 5.1.12.1

Multiply $e e$ .

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Step 5.1.12.1.1

Raise $e$ to the power of $1$ .

$\frac{\frac{e^{2} + 1}{e} \cdot \frac{e^{1} e - 1}{e}}{2}$

Step 5.1.12.1.2

Raise $e$ to the power of $1$ .

$\frac{\frac{e^{2} + 1}{e} \cdot \frac{e^{1} e^{1} - 1}{e}}{2}$

Step 5.1.12.1.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{e^{2} + 1}{e} \cdot \frac{e^{1 + 1} - 1}{e}}{2}$

Step 5.1.12.1.4

Add $1$ and $1$ .

$\frac{\frac{e^{2} + 1}{e} \cdot \frac{e^{2} - 1}{e}}{2}$

Step 5.1.12.2

Rewrite $e^{2} - 1$ in a factored form.

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Step 5.1.12.2.1

Rewrite $1$ as $1^{2}$ .

$\frac{\frac{e^{2} + 1}{e} \cdot \frac{e^{2} - 1^{2}}{e}}{2}$

Step 5.1.12.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = e$ and $b = 1$ .

$\frac{\frac{e^{2} + 1}{e} \cdot \frac{(e + 1) (e - 1)}{e}}{2}$

Step 5.2

Multiply $\frac{e^{2} + 1}{e}$ by $\frac{(e + 1) (e - 1)}{e}$ .

$\frac{\frac{(e^{2} + 1) ((e + 1) (e - 1))}{e e}}{2}$

Step 5.3

Simplify the denominator.

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Step 5.3.1

Raise $e$ to the power of $1$ .

$\frac{\frac{(e^{2} + 1) (e + 1) (e - 1)}{e^{1} e}}{2}$

Step 5.3.2

Raise $e$ to the power of $1$ .

$\frac{\frac{(e^{2} + 1) (e + 1) (e - 1)}{e^{1} e^{1}}}{2}$

Step 5.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\frac{(e^{2} + 1) (e + 1) (e - 1)}{e^{1 + 1}}}{2}$

Step 5.3.4

Add $1$ and $1$ .

$\frac{\frac{(e^{2} + 1) (e + 1) (e - 1)}{e^{2}}}{2}$

Step 5.4

Multiply the numerator by the reciprocal of the denominator.

$\frac{(e^{2} + 1) (e + 1) (e - 1)}{e^{2}} \cdot \frac{1}{2}$

Step 5.5

Combine.

$\frac{(e^{2} + 1) (e + 1) (e - 1) \cdot 1}{e^{2} \cdot 2}$

Step 5.6

Simplify the expression.

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Step 5.6.1

Multiply $e^{2} + 1$ by $1$ .

$\frac{(e^{2} + 1) (e + 1) (e - 1)}{e^{2} \cdot 2}$

Step 5.6.2

Move $2$ to the left of $e^{2}$ .

$\frac{(e^{2} + 1) (e + 1) (e - 1)}{2 e^{2}}$

Step 6