Calculus Examples

$y = 7 e^{x}$ , $y = 7 x e^{x}$ , $x = 0$

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.

$7 e^{x} = 7 x e^{x}$

Step 1.2

Solve $7 e^{x} = 7 x e^{x}$ for $x$ .

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Step 1.2.1

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

$\ln (7 e^{x}) = \ln (7 x e^{x})$

Step 1.2.2

Expand the left side.

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Step 1.2.2.1

Rewrite $\ln (7 e^{x})$ as $\ln (7) + \ln (e^{x})$ .

$\ln (7) + \ln (e^{x}) = \ln (7 x e^{x})$

Step 1.2.2.2

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

$\ln (7) + x \ln (e) = \ln (7 x e^{x})$

Step 1.2.2.3

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

$\ln (7) + x \cdot 1 = \ln (7 x e^{x})$

Step 1.2.2.4

Multiply $x$ by $1$ .

$\ln (7) + x = \ln (7 x e^{x})$

Step 1.2.3

Expand the right side.

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Step 1.2.3.1

Rewrite $\ln (7 x e^{x})$ as $\ln (7 x) + \ln (e^{x})$ .

$\ln (7) + x = \ln (7 x) + \ln (e^{x})$

Step 1.2.3.2

Rewrite $\ln (7 x)$ as $\ln (7) + \ln (x)$ .

$\ln (7) + x = \ln (7) + \ln (x) + \ln (e^{x})$

Step 1.2.3.3

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

$\ln (7) + x = \ln (7) + \ln (x) + x \ln (e)$

Step 1.2.3.4

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

$\ln (7) + x = \ln (7) + \ln (x) + x \cdot 1$

Step 1.2.3.5

Multiply $x$ by $1$ .

$\ln (7) + x = \ln (7) + \ln (x) + x$

Step 1.2.4

Simplify the right side.

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Step 1.2.4.1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\ln (7) + x = \ln (7 x) + x$

Step 1.2.5

Move all the terms containing a logarithm to the left side of the equation.

$\ln (7) - \ln (7 x) = - x + x$

Step 1.2.6

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\ln (\frac{7}{7 x}) = - x + x$

Step 1.2.7

Cancel the common factor of $7$ .

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Step 1.2.7.1

Cancel the common factor.

$\ln (\frac{7}{7 x}) = - x + x$

Step 1.2.7.2

Rewrite the expression.

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

Step 1.2.8

Add $- x$ and $x$ .

$\ln (\frac{1}{x}) = 0$

Step 1.2.9

To solve for $x$ , rewrite the equation using properties of logarithms.

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

Step 1.2.10

Rewrite $\ln (\frac{1}{x}) = 0$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{0} = \frac{1}{x}$

Step 1.2.11

Solve for $x$ .

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Step 1.2.11.1

Rewrite the equation as $\frac{1}{x} = e^{0}$ .

$\frac{1}{x} = e^{0}$

Step 1.2.11.2

Anything raised to $0$ is $1$ .

$\frac{1}{x} = 1$

Step 1.2.11.3

Find the LCD of the terms in the equation.

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Step 1.2.11.3.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x, 1 + y = 7 x e^{x}$

Step 1.2.11.3.2

The LCM of one and any expression is the expression.

$x + y = 7 x e^{x}$

Step 1.2.11.4

Multiply each term in $\frac{1}{x} = 1$ by $x$ to eliminate the fractions.

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Step 1.2.11.4.1

Multiply each term in $\frac{1}{x} = 1$ by $x$ .

$\frac{1}{x} x = 1 x$

Step 1.2.11.4.2

Simplify the left side.

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Step 1.2.11.4.2.1

Cancel the common factor of $x$ .

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Step 1.2.11.4.2.1.1

Cancel the common factor.

$\frac{1}{x} x = 1 x$

Step 1.2.11.4.2.1.2

Rewrite the expression.

$1 = 1 x$

Step 1.2.11.4.3

Simplify the right side.

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Step 1.2.11.4.3.1

Multiply $x$ by $1$ .

$1 = x$

Step 1.2.11.5

Rewrite the equation as $x = 1$ .

$x = 1$

Step 1.3

Evaluate $y$ when $x = 1$ .

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Step 1.3.1

Substitute $1$ for $x$ .

$y = 7 (1) \cdot e^{1}$

Step 1.3.2

Substitute $1$ for $x$ in $y = 7 (1) \cdot e^{1}$ and solve for $y$ .

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Step 1.3.2.1

Remove parentheses.

$y = 7 (1) \cdot e$

Step 1.3.2.2

Multiply $7$ by $1$ .

$y = 7 e$

Step 1.4

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

$(1, 7 e)$

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.

$A r e a = \int_{0}^{1} 7 x e^{x} d x - \int_{0}^{1} 7 e^{x} d x$

Step 3

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

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Step 3.1

Combine the integrals into a single integral.

$\int_{0}^{1} 7 x e^{x} - (7 e^{x}) d x$

Step 3.2

Multiply $7$ by $- 1$ .

$\int_{0}^{1} 7 x e^{x} - 7 e^{x} d x$

Step 3.3

Split the single integral into multiple integrals.

$\int_{0}^{1} 7 x e^{x} d x + \int_{0}^{1} - 7 e^{x} d x$

Step 3.4

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$7 \int_{0}^{1} x e^{x} d x + \int_{0}^{1} - 7 e^{x} d x$

Step 3.5

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

$7 (x e^{x}]_{0}^{1} - \int_{0}^{1} e^{x} d x) + \int_{0}^{1} - 7 e^{x} d x$

Step 3.6

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

$7 (x e^{x}]_{0}^{1} - (e^{x}]_{0}^{1})) + \int_{0}^{1} - 7 e^{x} d x$

Step 3.7

Since $- 7$ is constant with respect to $x$ , move $- 7$ out of the integral.

$7 (x e^{x}]_{0}^{1} - (e^{x}]_{0}^{1})) - 7 \int_{0}^{1} e^{x} d x$

Step 3.8

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

$7 (x e^{x}]_{0}^{1} - (e^{x}]_{0}^{1})) - 7 (e^{x}]_{0}^{1})$

Step 3.9

Substitute and simplify.

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Step 3.9.1

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

$7 ((1 e^{1}) + 0 e^{0} - (e^{x}]_{0}^{1})) - 7 (e^{x}]_{0}^{1})$

Step 3.9.2

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

$7 ((1 e^{1}) + 0 e^{0} - ((e^{1}) - e^{0})) - 7 (e^{x}]_{0}^{1})$

Step 3.9.3

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

$7 ((1 e^{1}) + 0 e^{0} - (e^{1} - e^{0})) - 7 (e^{1} - e^{0})$

Step 3.9.4

Simplify.

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Step 3.9.4.1

Simplify.

$7 (1 e + 0 e^{0} - (e^{1} - e^{0})) - 7 (e^{1} - e^{0})$

Step 3.9.4.2

Multiply $e$ by $1$ .

$7 (e + 0 e^{0} - (e^{1} - e^{0})) - 7 (e^{1} - e^{0})$

Step 3.9.4.3

Anything raised to $0$ is $1$ .

$7 (e + 0 \cdot 1 - (e^{1} - e^{0})) - 7 (e^{1} - e^{0})$

Step 3.9.4.4

Multiply $0$ by $1$ .

$7 (e + 0 - (e^{1} - e^{0})) - 7 (e^{1} - e^{0})$

Step 3.9.4.5

Add $e$ and $0$ .

$7 (e - (e^{1} - e^{0})) - 7 (e^{1} - e^{0})$

Step 3.9.4.6

Simplify.

$7 (e - (e - e^{0})) - 7 (e^{1} - e^{0})$

Step 3.9.4.7

Anything raised to $0$ is $1$ .

$7 (e - (e - 1 \cdot 1)) - 7 (e^{1} - e^{0})$

Step 3.9.4.8

Multiply $- 1$ by $1$ .

$7 (e - (e - 1)) - 7 (e^{1} - e^{0})$

Step 3.9.4.9

Simplify.

$7 (e - (e - 1)) - 7 (e - e^{0})$

Step 3.9.4.10

Anything raised to $0$ is $1$ .

$7 (e - (e - 1)) - 7 (e - 1 \cdot 1)$

Step 3.9.4.11

Multiply $- 1$ by $1$ .

$7 (e - (e - 1)) - 7 (e - 1)$

Step 3.10

Simplify.

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Step 3.10.1

Simplify each term.

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Step 3.10.1.1

Simplify each term.

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Step 3.10.1.1.1

Apply the distributive property.

$7 (e - e - - 1) - 7 (e - 1)$

Step 3.10.1.1.2

Multiply $- 1$ by $- 1$ .

$7 (e - e + 1) - 7 (e - 1)$

Step 3.10.1.2

Subtract $e$ from $e$ .

$7 (0 + 1) - 7 (e - 1)$

Step 3.10.1.3

Add $0$ and $1$ .

$7 \cdot 1 - 7 (e - 1)$

Step 3.10.1.4

Multiply $7$ by $1$ .

$7 - 7 (e - 1)$

Step 3.10.1.5

Apply the distributive property.

$7 - 7 e - 7 \cdot - 1$

Step 3.10.1.6

Multiply $- 7$ by $- 1$ .

$7 - 7 e + 7$

Step 3.10.2

Add $7$ and $7$ .

$14 - 7 e$

Step 4