Calculus Examples

$x = - 1$ , $x = 2$ , $y = 3 e^{3 x}$ , $y = 2 e^{3 x} + 1$

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.

$3 e^{3 x} = 2 e^{3 x} + 1$

Step 1.2

Solve $3 e^{3 x} = 2 e^{3 x} + 1$ for $x$ .

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Step 1.2.1

Move all terms containing $x$ to the left side of the equation.

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Step 1.2.1.1

Subtract $2 e^{3 x}$ from both sides of the equation.

$3 e^{3 x} - 2 e^{3 x} = 1$

Step 1.2.1.2

Subtract $2 e^{3 x}$ from $3 e^{3 x}$ .

$e^{3 x} = 1$

Step 1.2.2

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

$\ln (e^{3 x}) = \ln (1)$

Step 1.2.3

Expand the left side.

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Step 1.2.3.1

Expand $\ln (e^{3 x})$ by moving $3 x$ outside the logarithm.

$3 x \ln (e) = \ln (1)$

Step 1.2.3.2

The natural logarithm of $e$ is $1$ .

$3 x \cdot 1 = \ln (1)$

Step 1.2.3.3

Multiply $3$ by $1$ .

$3 x = \ln (1)$

Step 1.2.4

Simplify the right side.

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Step 1.2.4.1

The natural logarithm of $1$ is $0$ .

$3 x = 0$

Step 1.2.5

Divide each term in $3 x = 0$ by $3$ and simplify.

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Step 1.2.5.1

Divide each term in $3 x = 0$ by $3$ .

$\frac{3 x}{3} = \frac{0}{3}$

Step 1.2.5.2

Simplify the left side.

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Step 1.2.5.2.1

Cancel the common factor of $3$ .

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Step 1.2.5.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{0}{3}$

Step 1.2.5.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{3}$

Step 1.2.5.3

Simplify the right side.

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Step 1.2.5.3.1

Divide $0$ by $3$ .

$x = 0$

Step 1.3

Evaluate $y$ when $x = 0$ .

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Step 1.3.1

Substitute $0$ for $x$ .

$y = 2 e^{3 (0)} + 1$

Step 1.3.2

Simplify $2 e^{3 (0)} + 1$ .

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Step 1.3.2.1

Simplify each term.

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Step 1.3.2.1.1

Multiply $3$ by $0$ .

$y = 2 e^{0} + 1$

Step 1.3.2.1.2

Anything raised to $0$ is $1$ .

$y = 2 \cdot 1 + 1$

Step 1.3.2.1.3

Multiply $2$ by $1$ .

$y = 2 + 1$

Step 1.3.2.2

Add $2$ and $1$ .

$y = 3$

Step 1.4

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(0, 3)$

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.

$A r e a = \int_{- 1}^{0} 2 e^{3 x} + 1 d x - \int_{- 1}^{0} 3 e^{3 x} d x$

Step 3

Integrate to find the area between $- 1$ and $0$ .

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Step 3.1

Combine the integrals into a single integral.

$\int_{- 1}^{0} 2 e^{3 x} + 1 - (3 e^{3 x}) d x$

Step 3.2

Multiply $3$ by $- 1$ .

$\int_{- 1}^{0} 2 e^{3 x} + 1 - 3 e^{3 x} d x$

Step 3.3

Subtract $3 e^{3 x}$ from $2 e^{3 x}$ .

$\int_{- 1}^{0} 1 - e^{3 x} d x$

Step 3.4

Split the single integral into multiple integrals.

$\int_{- 1}^{0} d x + \int_{- 1}^{0} - e^{3 x} d x$

Step 3.5

Apply the constant rule.

$x]_{- 1}^{0} + \int_{- 1}^{0} - e^{3 x} d x$

Step 3.6

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

$x]_{- 1}^{0} - \int_{- 1}^{0} e^{3 x} d x$

Step 3.7

Let $u = 3 x$ . Then $d u = 3 d x$ , so $\frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.7.1

Let $u = 3 x$ . Find $\frac{d u}{d x}$ .

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Step 3.7.1.1

Differentiate $3 x$ .

$\frac{d}{d x} [3 x]$

Step 3.7.1.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x]$

Step 3.7.1.3

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

$3 \cdot 1$

Step 3.7.1.4

Multiply $3$ by $1$ .

$3$

Step 3.7.2

Substitute the lower limit in for $x$ in $u = 3 x$ .

$u_{lower} = 3 \cdot - 1$

Step 3.7.3

Multiply $3$ by $- 1$ .

$u_{lower} = - 3$

Step 3.7.4

Substitute the upper limit in for $x$ in $u = 3 x$ .

$u_{upper} = 3 \cdot 0$

Step 3.7.5

Multiply $3$ by $0$ .

$u_{upper} = 0$

Step 3.7.6

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

$u_{lower} = - 3$

$u_{upper} = 0$

Step 3.7.7

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

$x]_{- 1}^{0} - \int_{- 3}^{0} e^{u} \frac{1}{3} d u$

Step 3.8

Combine $e^{u}$ and $\frac{1}{3}$ .

$x]_{- 1}^{0} - \int_{- 3}^{0} \frac{e^{u}}{3} d u$

Step 3.9

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$x]_{- 1}^{0} - (\frac{1}{3} \int_{- 3}^{0} e^{u} d u)$

Step 3.10

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

$x]_{- 1}^{0} - \frac{1}{3} e^{u}]_{- 3}^{0}$

Step 3.11

Substitute and simplify.

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Step 3.11.1

Evaluate $x$ at $0$ and at $- 1$ .

$(0) + 1 - \frac{1}{3} e^{u}]_{- 3}^{0}$

Step 3.11.2

Evaluate $e^{u}$ at $0$ and at $- 3$ .

$(0) + 1 - \frac{1}{3} ((e^{0}) - e^{- 3})$

Step 3.11.3

Simplify.

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Step 3.11.3.1

Add $0$ and $1$ .

$1 - \frac{1}{3} ((e^{0}) - e^{- 3})$

Step 3.11.3.2

Anything raised to $0$ is $1$ .

$1 - \frac{1}{3} (1 - e^{- 3})$

Step 3.12

Simplify.

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Step 3.12.1

Simplify each term.

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Step 3.12.1.1

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

$1 - \frac{1}{3} (1 - \frac{1}{e^{3}})$

Step 3.12.1.2

Apply the distributive property.

$1 - \frac{1}{3} \cdot 1 - \frac{1}{3} (- \frac{1}{e^{3}})$

Step 3.12.1.3

Multiply $- 1$ by $1$ .

$1 - \frac{1}{3} - \frac{1}{3} (- \frac{1}{e^{3}})$

Step 3.12.1.4

Multiply $- \frac{1}{3} (- \frac{1}{e^{3}})$ .

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Step 3.12.1.4.1

Multiply $- 1$ by $- 1$ .

$1 - \frac{1}{3} + 1 (\frac{1}{3}) \frac{1}{e^{3}}$

Step 3.12.1.4.2

Multiply $\frac{1}{3}$ by $1$ .

$1 - \frac{1}{3} + \frac{1}{3} \cdot \frac{1}{e^{3}}$

Step 3.12.1.4.3

Multiply $\frac{1}{3}$ by $\frac{1}{e^{3}}$ .

$1 - \frac{1}{3} + \frac{1}{3 e^{3}}$

Step 3.12.2

Write $1$ as a fraction with a common denominator.

$\frac{3}{3} - \frac{1}{3} + \frac{1}{3 e^{3}}$

Step 3.12.3

Combine the numerators over the common denominator.

$\frac{3 - 1}{3} + \frac{1}{3 e^{3}}$

Step 3.12.4

Subtract $1$ from $3$ .

$\frac{2}{3} + \frac{1}{3 e^{3}}$

Step 4

$A r e a = \int_{0}^{2} 3 e^{3 x} d x - \int_{0}^{2} 2 e^{3 x} + 1 d x$

Step 5

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

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Step 5.1

Combine the integrals into a single integral.

$\int_{0}^{2} 3 e^{3 x} - (2 e^{3 x} + 1) d x$

Step 5.2

Simplify each term.

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Step 5.2.1

Apply the distributive property.

$3 e^{3 x} - (2 e^{3 x}) - 1 \cdot 1$

Step 5.2.2

Multiply $2$ by $- 1$ .

$3 e^{3 x} - 2 e^{3 x} - 1 \cdot 1$

Step 5.2.3

Multiply $- 1$ by $1$ .

$3 e^{3 x} - 2 e^{3 x} - 1$

$\int_{0}^{2} 3 e^{3 x} - 2 e^{3 x} - 1 d x$

Step 5.3

Subtract $2 e^{3 x}$ from $3 e^{3 x}$ .

$\int_{0}^{2} e^{3 x} - 1 d x$

Step 5.4

Split the single integral into multiple integrals.

$\int_{0}^{2} e^{3 x} d x + \int_{0}^{2} - 1 d x$

Step 5.5

Let $u = 3 x$ . Then $d u = 3 d x$ , so $\frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.5.1

Let $u = 3 x$ . Find $\frac{d u}{d x}$ .

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Step 5.5.1.1

Differentiate $3 x$ .

$\frac{d}{d x} [3 x]$

Step 5.5.1.2

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x]$

Step 5.5.1.3

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

$3 \cdot 1$

Step 5.5.1.4

Multiply $3$ by $1$ .

$3$

Step 5.5.2

Substitute the lower limit in for $x$ in $u = 3 x$ .

$u_{lower} = 3 \cdot 0$

Step 5.5.3

Multiply $3$ by $0$ .

$u_{lower} = 0$

Step 5.5.4

Substitute the upper limit in for $x$ in $u = 3 x$ .

$u_{upper} = 3 \cdot 2$

Step 5.5.5

Multiply $3$ by $2$ .

$u_{upper} = 6$

Step 5.5.6

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

$u_{lower} = 0$

$u_{upper} = 6$

Step 5.5.7

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

$\int_{0}^{6} e^{u} \frac{1}{3} d u + \int_{0}^{2} - 1 d x$

Step 5.6

Combine $e^{u}$ and $\frac{1}{3}$ .

$\int_{0}^{6} \frac{e^{u}}{3} d u + \int_{0}^{2} - 1 d x$

Step 5.7

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{3} \int_{0}^{6} e^{u} d u + \int_{0}^{2} - 1 d x$

Step 5.8

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

$\frac{1}{3} e^{u}]_{0}^{6} + \int_{0}^{2} - 1 d x$

Step 5.9

Apply the constant rule.

$\frac{1}{3} e^{u}]_{0}^{6} + - x]_{0}^{2}$

Step 5.10

Substitute and simplify.

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Step 5.10.1

Evaluate $e^{u}$ at $6$ and at $0$ .

$\frac{1}{3} ((e^{6}) - e^{0}) + - x]_{0}^{2}$

Step 5.10.2

Evaluate $- x$ at $2$ and at $0$ .

$\frac{1}{3} ((e^{6}) - e^{0}) + (- 1 \cdot 2) + 0$

Step 5.10.3

Simplify.

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Step 5.10.3.1

Anything raised to $0$ is $1$ .

$\frac{1}{3} (e^{6} - 1 \cdot 1) + (- 1 \cdot 2) + 0$

Step 5.10.3.2

Multiply $- 1$ by $1$ .

$\frac{1}{3} (e^{6} - 1) + (- 1 \cdot 2) + 0$

Step 5.10.3.3

Multiply $- 1$ by $2$ .

$\frac{1}{3} (e^{6} - 1) - 2 + 0$

Step 5.10.3.4

Add $- 2$ and $0$ .

$\frac{1}{3} (e^{6} - 1) - 2$

Step 5.11

Simplify.

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Step 5.11.1

Simplify each term.

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Step 5.11.1.1

Apply the distributive property.

$\frac{1}{3} e^{6} + \frac{1}{3} \cdot - 1 - 2$

Step 5.11.1.2

Combine $\frac{1}{3}$ and $e^{6}$ .

$\frac{e^{6}}{3} + \frac{1}{3} \cdot - 1 - 2$

Step 5.11.1.3

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

$\frac{e^{6}}{3} + \frac{- 1}{3} - 2$

Step 5.11.1.4

Move the negative in front of the fraction.

$\frac{e^{6}}{3} - \frac{1}{3} - 2$

Step 5.11.2

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{e^{6}}{3} - \frac{1}{3} - 2 \cdot \frac{3}{3}$

Step 5.11.3

Combine $- 2$ and $\frac{3}{3}$ .

$\frac{e^{6}}{3} - \frac{1}{3} + \frac{- 2 \cdot 3}{3}$

Step 5.11.4

Combine the numerators over the common denominator.

$\frac{e^{6}}{3} + \frac{- 1 - 2 \cdot 3}{3}$

Step 5.11.5

Simplify the numerator.

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Step 5.11.5.1

Multiply $- 2$ by $3$ .

$\frac{e^{6}}{3} + \frac{- 1 - 6}{3}$

Step 5.11.5.2

Subtract $6$ from $- 1$ .

$\frac{e^{6}}{3} + \frac{- 7}{3}$

Step 5.11.6

Move the negative in front of the fraction.

$\frac{e^{6}}{3} - \frac{7}{3}$

Step 6

Add the areas $\frac{2}{3} + \frac{1}{3 e^{3}} + \frac{e^{6}}{3} - \frac{7}{3}$ .

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Step 6.1

Combine the numerators over the common denominator.

$\frac{2 + e^{6} - 7}{3} + \frac{1}{3 e^{3}}$

Step 6.2

Subtract $7$ from $2$ .

$\frac{e^{6} - 5}{3} + \frac{1}{3 e^{3}}$

Step 6.3

To write $\frac{e^{6} - 5}{3}$ as a fraction with a common denominator, multiply by $\frac{e^{3}}{e^{3}}$ .

$\frac{e^{6} - 5}{3} \cdot \frac{e^{3}}{e^{3}} + \frac{1}{3 e^{3}}$

Step 6.4

Combine fractions.

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Step 6.4.1

Multiply $\frac{e^{6} - 5}{3}$ by $\frac{e^{3}}{e^{3}}$ .

$\frac{(e^{6} - 5) e^{3}}{3 e^{3}} + \frac{1}{3 e^{3}}$

Step 6.4.2

Combine the numerators over the common denominator.

$\frac{(e^{6} - 5) e^{3} + 1}{3 e^{3}}$

Step 6.5

Simplify the numerator.

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Step 6.5.1

Apply the distributive property.

$\frac{e^{6} e^{3} - 5 e^{3} + 1}{3 e^{3}}$

Step 6.5.2

Multiply $e^{6}$ by $e^{3}$ by adding the exponents.

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Step 6.5.2.1

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

$\frac{e^{6 + 3} - 5 e^{3} + 1}{3 e^{3}}$

Step 6.5.2.2

Add $6$ and $3$ .

$\frac{e^{9} - 5 e^{3} + 1}{3 e^{3}}$

Step 7