Calculus Examples

$f (x) = 8 + 4 e^{0.5 x}$ , $[- 3, 3]$

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.

$8 + 4 e^{0.5 x} = 0$

Step 1.2

Solve $8 + 4 e^{0.5 x} = 0$ for $x$ .

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Step 1.2.1

Subtract $8$ from both sides of the equation.

$4 e^{0.5 x} = - 8$

Step 1.2.2

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (4 e^{0.5 x}) = \ln (- 8)$

Step 1.2.3

The equation cannot be solved because $\ln (- 8)$ is undefined.

Undefined

Step 1.2.4

There is no solution for $4 e^{0.5 x} = - 8$

No solution

Step 2

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.

$A r e a = \int_{- 3}^{3} 8 + 4 e^{0.5 x} d x - \int_{- 3}^{3} 0 d x$

Step 3

Integrate to find the area between $- 3$ and $3$ .

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Step 3.1

Combine the integrals into a single integral.

$\int_{- 3}^{3} 8 + 4 e^{0.5 x} - (0) d x$

Step 3.2

Subtract $0$ from $8 + 4 e^{0.5 x}$ .

$\int_{- 3}^{3} 8 + 4 e^{0.5 x} d x$

Step 3.3

Split the single integral into multiple integrals.

$\int_{- 3}^{3} 8 d x + \int_{- 3}^{3} 4 e^{0.5 x} d x$

Step 3.4

Apply the constant rule.

$8 x]_{- 3}^{3} + \int_{- 3}^{3} 4 e^{0.5 x} d x$

Step 3.5

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$8 x]_{- 3}^{3} + 4 \int_{- 3}^{3} e^{0.5 x} d x$

Step 3.6

Let $u = 0.5 x$ . Then $d u = 0.5 d x$ , so $\frac{1}{0.5} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.6.1

Let $u = 0.5 x$ . Find $\frac{d u}{d x}$ .

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Step 3.6.1.1

Differentiate $0.5 x$ .

$\frac{d}{d x} [0.5 x]$

Step 3.6.1.2

Since $0.5$ is constant with respect to $x$ , the derivative of $0.5 x$ with respect to $x$ is $0.5 \frac{d}{d x} [x]$ .

$0.5 \frac{d}{d x} [x]$

Step 3.6.1.3

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

$0.5 \cdot 1$

Step 3.6.1.4

Multiply $0.5$ by $1$ .

$0.5$

Step 3.6.2

Substitute the lower limit in for $x$ in $u = 0.5 x$ .

$u_{lower} = 0.5 \cdot - 3$

Step 3.6.3

Multiply $0.5$ by $- 3$ .

$u_{lower} = - 1.5$

Step 3.6.4

Substitute the upper limit in for $x$ in $u = 0.5 x$ .

$u_{upper} = 0.5 \cdot 3$

Step 3.6.5

Multiply $0.5$ by $3$ .

$u_{upper} = 1.5$

Step 3.6.6

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

$u_{lower} = - 1.5$

$u_{upper} = 1.5$

Step 3.6.7

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

$8 x]_{- 3}^{3} + 4 \int_{- 1.5}^{1.5} e^{u} \frac{1}{0.5} d u$

Step 3.7

Combine $e^{u}$ and $\frac{1}{0.5}$ .

$8 x]_{- 3}^{3} + 4 \int_{- 1.5}^{1.5} \frac{e^{u}}{0.5} d u$

Step 3.8

Since $\frac{1}{0.5}$ is constant with respect to $u$ , move $\frac{1}{0.5}$ out of the integral.

$8 x]_{- 3}^{3} + 4 (\frac{1}{0.5} \int_{- 1.5}^{1.5} e^{u} d u)$

Step 3.9

Combine $\frac{1}{0.5}$ and $4$ .

$8 x]_{- 3}^{3} + \frac{4}{0.5} \int_{- 1.5}^{1.5} e^{u} d u$

Step 3.10

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

$8 x]_{- 3}^{3} + \frac{4}{0.5} e^{u}]_{- 1.5}^{1.5}$

Step 3.11

Substitute and simplify.

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Step 3.11.1

Evaluate $8 x$ at $3$ and at $- 3$ .

$(8 \cdot 3) - 8 \cdot - 3 + \frac{4}{0.5} e^{u}]_{- 1.5}^{1.5}$

Step 3.11.2

Evaluate $e^{u}$ at $1.5$ and at $- 1.5$ .

$(8 \cdot 3) - 8 \cdot - 3 + \frac{4}{0.5} ((e^{1.5}) - e^{- 1.5})$

Step 3.11.3

Simplify.

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Step 3.11.3.1

Multiply $8$ by $3$ .

$24 - 8 \cdot - 3 + \frac{4}{0.5} ((e^{1.5}) - e^{- 1.5})$

Step 3.11.3.2

Multiply $- 8$ by $- 3$ .

$24 + 24 + \frac{4}{0.5} ((e^{1.5}) - e^{- 1.5})$

Step 3.11.3.3

Add $24$ and $24$ .

$48 + \frac{4}{0.5} (e^{1.5} - e^{- 1.5})$

Step 3.12

Simplify each term.

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Step 3.12.1

Divide $4$ by $0.5$ .

$48 + 8 (e^{1.5} - e^{- 1.5})$

Step 3.12.2

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

$48 + 8 (e^{1.5} - \frac{1}{e^{1.5}})$

Step 3.12.3

Apply the distributive property.

$48 + 8 e^{1.5} + 8 (- \frac{1}{e^{1.5}})$

Step 3.12.4

Multiply $8 (- \frac{1}{e^{1.5}})$ .

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Step 3.12.4.1

Multiply $- 1$ by $8$ .

$48 + 8 e^{1.5} - 8 \frac{1}{e^{1.5}}$

Step 3.12.4.2

Combine $- 8$ and $\frac{1}{e^{1.5}}$ .

$48 + 8 e^{1.5} + \frac{- 8}{e^{1.5}}$

Step 3.12.5

Move the negative in front of the fraction.

$48 + 8 e^{1.5} - \frac{8}{e^{1.5}}$

Step 4