Calculus Examples

$f (x) = 6 x - 12$ ; $[1, 5]$

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.

$6 x - 12 = 0$

Step 1.2

Solve $6 x - 12 = 0$ for $x$ .

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Step 1.2.1

Add $12$ to both sides of the equation.

$6 x = 12$

Step 1.2.2

Divide each term in $6 x = 12$ by $6$ and simplify.

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Step 1.2.2.1

Divide each term in $6 x = 12$ by $6$ .

$\frac{6 x}{6} = \frac{12}{6}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Cancel the common factor of $6$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

$\frac{6 x}{6} = \frac{12}{6}$

Step 1.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{12}{6}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Divide $12$ by $6$ .

$x = 2$

Step 1.3

Substitute $2$ for $x$ .

$y = 0$

Step 1.4

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

$(2, 0)$

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.

$A r e a = \int_{1}^{2} 0 d x - \int_{1}^{2} 6 x - 12 d x$

Step 3

Integrate to find the area between $1$ and $2$ .

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Step 3.1

Combine the integrals into a single integral.

$\int_{1}^{2} 0 - (6 x - 12) d x$

Step 3.2

Subtract $6 x - 12$ from $0$ .

$\int_{1}^{2} - (6 x - 12) d x$

Step 3.3

Apply the distributive property.

$\int_{1}^{2} - (6 x) + 12 d x$

Step 3.4

Multiply.

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Step 3.4.1

Multiply $6$ by $- 1$ .

$- 6 x - - 12$

Step 3.4.2

Multiply $- 1$ by $- 12$ .

$- 6 x + 12$

$\int_{1}^{2} - 6 x + 12 d x$

Step 3.5

Split the single integral into multiple integrals.

$\int_{1}^{2} - 6 x d x + \int_{1}^{2} 12 d x$

Step 3.6

Since $- 6$ is constant with respect to $x$ , move $- 6$ out of the integral.

$- 6 \int_{1}^{2} x d x + \int_{1}^{2} 12 d x$

Step 3.7

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$- 6 (\frac{1}{2} x^{2}]_{1}^{2}) + \int_{1}^{2} 12 d x$

Step 3.8

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

$- 6 (\frac{x^{2}}{2}]_{1}^{2}) + \int_{1}^{2} 12 d x$

Step 3.9

Apply the constant rule.

$- 6 (\frac{x^{2}}{2}]_{1}^{2}) + 12 x]_{1}^{2}$

Step 3.10

Substitute and simplify.

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Step 3.10.1

Evaluate $\frac{x^{2}}{2}$ at $2$ and at $1$ .

$- 6 ((\frac{2^{2}}{2}) - \frac{1^{2}}{2}) + 12 x]_{1}^{2}$

Step 3.10.2

Evaluate $12 x$ at $2$ and at $1$ .

$- 6 (\frac{2^{2}}{2} - \frac{1^{2}}{2}) + 12 \cdot 2 - 12 \cdot 1$

Step 3.10.3

Simplify.

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Step 3.10.3.1

Raise $2$ to the power of $2$ .

$- 6 (\frac{4}{2} - \frac{1^{2}}{2}) + 12 \cdot 2 - 12 \cdot 1$

Step 3.10.3.2

Cancel the common factor of $4$ and $2$ .

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Step 3.10.3.2.1

Factor $2$ out of $4$ .

$- 6 (\frac{2 \cdot 2}{2} - \frac{1^{2}}{2}) + 12 \cdot 2 - 12 \cdot 1$

Step 3.10.3.2.2

Cancel the common factors.

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Step 3.10.3.2.2.1

Factor $2$ out of $2$ .

$- 6 (\frac{2 \cdot 2}{2 (1)} - \frac{1^{2}}{2}) + 12 \cdot 2 - 12 \cdot 1$

Step 3.10.3.2.2.2

Cancel the common factor.

$- 6 (\frac{2 \cdot 2}{2 \cdot 1} - \frac{1^{2}}{2}) + 12 \cdot 2 - 12 \cdot 1$

Step 3.10.3.2.2.3

Rewrite the expression.

$- 6 (\frac{2}{1} - \frac{1^{2}}{2}) + 12 \cdot 2 - 12 \cdot 1$

Step 3.10.3.2.2.4

Divide $2$ by $1$ .

$- 6 (2 - \frac{1^{2}}{2}) + 12 \cdot 2 - 12 \cdot 1$

Step 3.10.3.3

One to any power is one.

$- 6 (2 - \frac{1}{2}) + 12 \cdot 2 - 12 \cdot 1$

Step 3.10.3.4

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

$- 6 (2 \cdot \frac{2}{2} - \frac{1}{2}) + 12 \cdot 2 - 12 \cdot 1$

Step 3.10.3.5

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

$- 6 (\frac{2 \cdot 2}{2} - \frac{1}{2}) + 12 \cdot 2 - 12 \cdot 1$

Step 3.10.3.6

Combine the numerators over the common denominator.

$- 6 \frac{2 \cdot 2 - 1}{2} + 12 \cdot 2 - 12 \cdot 1$

Step 3.10.3.7

Simplify the numerator.

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Step 3.10.3.7.1

Multiply $2$ by $2$ .

$- 6 \frac{4 - 1}{2} + 12 \cdot 2 - 12 \cdot 1$

Step 3.10.3.7.2

Subtract $1$ from $4$ .

$- 6 (\frac{3}{2}) + 12 \cdot 2 - 12 \cdot 1$

Step 3.10.3.8

Combine $- 6$ and $\frac{3}{2}$ .

$\frac{- 6 \cdot 3}{2} + 12 \cdot 2 - 12 \cdot 1$

Step 3.10.3.9

Multiply $- 6$ by $3$ .

$\frac{- 18}{2} + 12 \cdot 2 - 12 \cdot 1$

Step 3.10.3.10

Cancel the common factor of $- 18$ and $2$ .

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Step 3.10.3.10.1

Factor $2$ out of $- 18$ .

$\frac{2 \cdot - 9}{2} + 12 \cdot 2 - 12 \cdot 1$

Step 3.10.3.10.2

Cancel the common factors.

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Step 3.10.3.10.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot - 9}{2 (1)} + 12 \cdot 2 - 12 \cdot 1$

Step 3.10.3.10.2.2

Cancel the common factor.

$\frac{2 \cdot - 9}{2 \cdot 1} + 12 \cdot 2 - 12 \cdot 1$

Step 3.10.3.10.2.3

Rewrite the expression.

$\frac{- 9}{1} + 12 \cdot 2 - 12 \cdot 1$

Step 3.10.3.10.2.4

Divide $- 9$ by $1$ .

$- 9 + 12 \cdot 2 - 12 \cdot 1$

Step 3.10.3.11

Multiply $12$ by $2$ .

$- 9 + 24 - 12 \cdot 1$

Step 3.10.3.12

Multiply $- 12$ by $1$ .

$- 9 + 24 - 12$

Step 3.10.3.13

Subtract $12$ from $24$ .

$- 9 + 12$

Step 3.10.3.14

Add $- 9$ and $12$ .

$3$

Step 4

$A r e a = \int_{2}^{5} 6 x - 12 d x - \int_{2}^{5} 0 d x$

Step 5

Integrate to find the area between $2$ and $5$ .

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Step 5.1

Combine the integrals into a single integral.

$\int_{2}^{5} 6 x - 12 - (0) d x$

Step 5.2

Subtract $0$ from $6 x - 12$ .

$\int_{2}^{5} 6 x - 12 d x$

Step 5.3

Split the single integral into multiple integrals.

$\int_{2}^{5} 6 x d x + \int_{2}^{5} - 12 d x$

Step 5.4

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$6 \int_{2}^{5} x d x + \int_{2}^{5} - 12 d x$

Step 5.5

By the Power Rule, the integral of $x$ with respect to $x$ is $\frac{1}{2} x^{2}$ .

$6 (\frac{1}{2} x^{2}]_{2}^{5}) + \int_{2}^{5} - 12 d x$

Step 5.6

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

$6 (\frac{x^{2}}{2}]_{2}^{5}) + \int_{2}^{5} - 12 d x$

Step 5.7

Apply the constant rule.

$6 (\frac{x^{2}}{2}]_{2}^{5}) + - 12 x]_{2}^{5}$

Step 5.8

Substitute and simplify.

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Step 5.8.1

Evaluate $\frac{x^{2}}{2}$ at $5$ and at $2$ .

$6 ((\frac{5^{2}}{2}) - \frac{2^{2}}{2}) + - 12 x]_{2}^{5}$

Step 5.8.2

Evaluate $- 12 x$ at $5$ and at $2$ .

$6 (\frac{5^{2}}{2} - \frac{2^{2}}{2}) + - 12 \cdot 5 + 12 \cdot 2$

Step 5.8.3

Simplify.

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Step 5.8.3.1

Raise $5$ to the power of $2$ .

$6 (\frac{25}{2} - \frac{2^{2}}{2}) - 12 \cdot 5 + 12 \cdot 2$

Step 5.8.3.2

Raise $2$ to the power of $2$ .

$6 (\frac{25}{2} - \frac{4}{2}) - 12 \cdot 5 + 12 \cdot 2$

Step 5.8.3.3

Cancel the common factor of $4$ and $2$ .

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Step 5.8.3.3.1

Factor $2$ out of $4$ .

$6 (\frac{25}{2} - \frac{2 \cdot 2}{2}) - 12 \cdot 5 + 12 \cdot 2$

Step 5.8.3.3.2

Cancel the common factors.

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Step 5.8.3.3.2.1

Factor $2$ out of $2$ .

$6 (\frac{25}{2} - \frac{2 \cdot 2}{2 (1)}) - 12 \cdot 5 + 12 \cdot 2$

Step 5.8.3.3.2.2

Cancel the common factor.

$6 (\frac{25}{2} - \frac{2 \cdot 2}{2 \cdot 1}) - 12 \cdot 5 + 12 \cdot 2$

Step 5.8.3.3.2.3

Rewrite the expression.

$6 (\frac{25}{2} - \frac{2}{1}) - 12 \cdot 5 + 12 \cdot 2$

Step 5.8.3.3.2.4

Divide $2$ by $1$ .

$6 (\frac{25}{2} - 1 \cdot 2) - 12 \cdot 5 + 12 \cdot 2$

Step 5.8.3.4

Multiply $- 1$ by $2$ .

$6 (\frac{25}{2} - 2) - 12 \cdot 5 + 12 \cdot 2$

Step 5.8.3.5

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

$6 (\frac{25}{2} - 2 \cdot \frac{2}{2}) - 12 \cdot 5 + 12 \cdot 2$

Step 5.8.3.6

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

$6 (\frac{25}{2} + \frac{- 2 \cdot 2}{2}) - 12 \cdot 5 + 12 \cdot 2$

Step 5.8.3.7

Combine the numerators over the common denominator.

$6 \frac{25 - 2 \cdot 2}{2} - 12 \cdot 5 + 12 \cdot 2$

Step 5.8.3.8

Simplify the numerator.

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Step 5.8.3.8.1

Multiply $- 2$ by $2$ .

$6 \frac{25 - 4}{2} - 12 \cdot 5 + 12 \cdot 2$

Step 5.8.3.8.2

Subtract $4$ from $25$ .

$6 (\frac{21}{2}) - 12 \cdot 5 + 12 \cdot 2$

Step 5.8.3.9

Combine $6$ and $\frac{21}{2}$ .

$\frac{6 \cdot 21}{2} - 12 \cdot 5 + 12 \cdot 2$

Step 5.8.3.10

Multiply $6$ by $21$ .

$\frac{126}{2} - 12 \cdot 5 + 12 \cdot 2$

Step 5.8.3.11

Cancel the common factor of $126$ and $2$ .

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Step 5.8.3.11.1

Factor $2$ out of $126$ .

$\frac{2 \cdot 63}{2} - 12 \cdot 5 + 12 \cdot 2$

Step 5.8.3.11.2

Cancel the common factors.

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Step 5.8.3.11.2.1

Factor $2$ out of $2$ .

$\frac{2 \cdot 63}{2 (1)} - 12 \cdot 5 + 12 \cdot 2$

Step 5.8.3.11.2.2

Cancel the common factor.

$\frac{2 \cdot 63}{2 \cdot 1} - 12 \cdot 5 + 12 \cdot 2$

Step 5.8.3.11.2.3

Rewrite the expression.

$\frac{63}{1} - 12 \cdot 5 + 12 \cdot 2$

Step 5.8.3.11.2.4

Divide $63$ by $1$ .

$63 - 12 \cdot 5 + 12 \cdot 2$

Step 5.8.3.12

Multiply $- 12$ by $5$ .

$63 - 60 + 12 \cdot 2$

Step 5.8.3.13

Multiply $12$ by $2$ .

$63 - 60 + 24$

Step 5.8.3.14

Add $- 60$ and $24$ .

$63 - 36$

Step 5.8.3.15

Subtract $36$ from $63$ .

$27$

Step 6

Add $3$ and $27$ .

$30$

Step 7