Calculus Examples

$y = x^{2} + 3$ ; $y = 6 x - 6$ ; $- 1 \leq x \leq 2$

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.

$x^{2} + 3 = 6 x - 6$

Step 1.2

Solve $x^{2} + 3 = 6 x - 6$ for $x$ .

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Step 1.2.1

Subtract $6 x$ from both sides of the equation.

$x^{2} + 3 - 6 x = - 6$

Step 1.2.2

Add $6$ to both sides of the equation.

$x^{2} + 3 - 6 x + 6 = 0$

Step 1.2.3

Add $3$ and $6$ .

$x^{2} - 6 x + 9 = 0$

Step 1.2.4

Factor using the perfect square rule.

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Step 1.2.4.1

Rewrite $9$ as $3^{2}$ .

$x^{2} - 6 x + 3^{2} = 0$

Step 1.2.4.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$6 x = 2 \cdot x \cdot 3$

Step 1.2.4.3

Rewrite the polynomial.

$x^{2} - 2 \cdot x \cdot 3 + 3^{2} = 0$

Step 1.2.4.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = x$ and $b = 3$ .

${(x - 3)}^{2} = 0$

Step 1.2.5

Set the $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 1.2.6

Add $3$ to both sides of the equation.

$x = 3$

Step 1.3

Evaluate $y$ when $x = 3$ .

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Step 1.3.1

Substitute $3$ for $x$ .

$y = 6 (3) - 6$

Step 1.3.2

Simplify $6 (3) - 6$ .

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Step 1.3.2.1

Multiply $6$ by $3$ .

$y = 18 - 6$

Step 1.3.2.2

Subtract $6$ from $18$ .

$y = 12$

Step 1.4

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

$(3, 12)$

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.

$A r e a = \int_{- 1}^{2} x^{2} + 3 d x - \int_{- 1}^{2} 6 x - 6 d x$

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.

$\int_{- 1}^{2} x^{2} + 3 - (6 x - 6) d x$

Step 3.2

Simplify each term.

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Step 3.2.1

Apply the distributive property.

$x^{2} + 3 - (6 x) - - 6$

Step 3.2.2

Multiply $6$ by $- 1$ .

$x^{2} + 3 - 6 x - - 6$

Step 3.2.3

Multiply $- 1$ by $- 6$ .

$x^{2} + 3 - 6 x + 6$

$\int_{- 1}^{2} x^{2} + 3 - 6 x + 6 d x$

Step 3.3

Add $3$ and $6$ .

$\int_{- 1}^{2} x^{2} - 6 x + 9 d x$

Step 3.4

Split the single integral into multiple integrals.

$\int_{- 1}^{2} x^{2} d x + \int_{- 1}^{2} - 6 x d x + \int_{- 1}^{2} 9 d x$

Step 3.5

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$\frac{1}{3} x^{3}]_{- 1}^{2} + \int_{- 1}^{2} - 6 x d x + \int_{- 1}^{2} 9 d x$

Step 3.6

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

$\frac{1}{3} x^{3}]_{- 1}^{2} - 6 \int_{- 1}^{2} x d x + \int_{- 1}^{2} 9 d x$

Step 3.7

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

$\frac{1}{3} x^{3}]_{- 1}^{2} - 6 (\frac{1}{2} x^{2}]_{- 1}^{2}) + \int_{- 1}^{2} 9 d x$

Step 3.8

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

$\frac{1}{3} x^{3}]_{- 1}^{2} - 6 (\frac{x^{2}}{2}]_{- 1}^{2}) + \int_{- 1}^{2} 9 d x$

Step 3.9

Apply the constant rule.

$\frac{1}{3} x^{3}]_{- 1}^{2} - 6 (\frac{x^{2}}{2}]_{- 1}^{2}) + 9 x]_{- 1}^{2}$

Step 3.10

Simplify the answer.

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Step 3.10.1

Combine $\frac{1}{3} x^{3}]_{- 1}^{2}$ and $9 x]_{- 1}^{2}$ .

$\frac{1}{3} x^{3} + 9 x]_{- 1}^{2} - 6 (\frac{x^{2}}{2}]_{- 1}^{2})$

Step 3.10.2

Substitute and simplify.

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Step 3.10.2.1

Evaluate $\frac{1}{3} x^{3} + 9 x$ at $2$ and at $- 1$ .

$(\frac{1}{3} \cdot 2^{3} + 9 \cdot 2) - (\frac{1}{3} {(- 1)}^{3} + 9 \cdot - 1) - 6 (\frac{x^{2}}{2}]_{- 1}^{2})$

Step 3.10.2.2

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

$(\frac{1}{3} \cdot 2^{3} + 9 \cdot 2) - (\frac{1}{3} {(- 1)}^{3} + 9 \cdot - 1) - 6 ((\frac{2^{2}}{2}) - \frac{{(- 1)}^{2}}{2})$

Step 3.10.2.3

Simplify.

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Step 3.10.2.3.1

Raise $2$ to the power of $3$ .

$\frac{1}{3} \cdot 8 + 9 \cdot 2 - (\frac{1}{3} {(- 1)}^{3} + 9 \cdot - 1) - 6 ((\frac{2^{2}}{2}) - \frac{{(- 1)}^{2}}{2})$

Step 3.10.2.3.2

Combine $\frac{1}{3}$ and $8$ .

$\frac{8}{3} + 9 \cdot 2 - (\frac{1}{3} {(- 1)}^{3} + 9 \cdot - 1) - 6 ((\frac{2^{2}}{2}) - \frac{{(- 1)}^{2}}{2})$

Step 3.10.2.3.3

Multiply $9$ by $2$ .

$\frac{8}{3} + 18 - (\frac{1}{3} {(- 1)}^{3} + 9 \cdot - 1) - 6 ((\frac{2^{2}}{2}) - \frac{{(- 1)}^{2}}{2})$

Step 3.10.2.3.4

To write $18$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{8}{3} + 18 \cdot \frac{3}{3} - (\frac{1}{3} {(- 1)}^{3} + 9 \cdot - 1) - 6 ((\frac{2^{2}}{2}) - \frac{{(- 1)}^{2}}{2})$

Step 3.10.2.3.5

Combine $18$ and $\frac{3}{3}$ .

$\frac{8}{3} + \frac{18 \cdot 3}{3} - (\frac{1}{3} {(- 1)}^{3} + 9 \cdot - 1) - 6 ((\frac{2^{2}}{2}) - \frac{{(- 1)}^{2}}{2})$

Step 3.10.2.3.6

Combine the numerators over the common denominator.

$\frac{8 + 18 \cdot 3}{3} - (\frac{1}{3} {(- 1)}^{3} + 9 \cdot - 1) - 6 ((\frac{2^{2}}{2}) - \frac{{(- 1)}^{2}}{2})$

Step 3.10.2.3.7

Simplify the numerator.

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Step 3.10.2.3.7.1

Multiply $18$ by $3$ .

$\frac{8 + 54}{3} - (\frac{1}{3} {(- 1)}^{3} + 9 \cdot - 1) - 6 ((\frac{2^{2}}{2}) - \frac{{(- 1)}^{2}}{2})$

Step 3.10.2.3.7.2

Add $8$ and $54$ .

$\frac{62}{3} - (\frac{1}{3} {(- 1)}^{3} + 9 \cdot - 1) - 6 ((\frac{2^{2}}{2}) - \frac{{(- 1)}^{2}}{2})$

Step 3.10.2.3.8

Raise $- 1$ to the power of $3$ .

$\frac{62}{3} - (\frac{1}{3} \cdot - 1 + 9 \cdot - 1) - 6 ((\frac{2^{2}}{2}) - \frac{{(- 1)}^{2}}{2})$

Step 3.10.2.3.9

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

$\frac{62}{3} - (\frac{- 1}{3} + 9 \cdot - 1) - 6 ((\frac{2^{2}}{2}) - \frac{{(- 1)}^{2}}{2})$

Step 3.10.2.3.10

Move the negative in front of the fraction.

$\frac{62}{3} - (- \frac{1}{3} + 9 \cdot - 1) - 6 ((\frac{2^{2}}{2}) - \frac{{(- 1)}^{2}}{2})$

Step 3.10.2.3.11

Multiply $9$ by $- 1$ .

$\frac{62}{3} - (- \frac{1}{3} - 9) - 6 ((\frac{2^{2}}{2}) - \frac{{(- 1)}^{2}}{2})$

Step 3.10.2.3.12

To write $- 9$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{62}{3} - (- \frac{1}{3} - 9 \cdot \frac{3}{3}) - 6 ((\frac{2^{2}}{2}) - \frac{{(- 1)}^{2}}{2})$

Step 3.10.2.3.13

Combine $- 9$ and $\frac{3}{3}$ .

$\frac{62}{3} - (- \frac{1}{3} + \frac{- 9 \cdot 3}{3}) - 6 ((\frac{2^{2}}{2}) - \frac{{(- 1)}^{2}}{2})$

Step 3.10.2.3.14

Combine the numerators over the common denominator.

$\frac{62}{3} - \frac{- 1 - 9 \cdot 3}{3} - 6 ((\frac{2^{2}}{2}) - \frac{{(- 1)}^{2}}{2})$

Step 3.10.2.3.15

Simplify the numerator.

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Step 3.10.2.3.15.1

Multiply $- 9$ by $3$ .

$\frac{62}{3} - \frac{- 1 - 27}{3} - 6 ((\frac{2^{2}}{2}) - \frac{{(- 1)}^{2}}{2})$

Step 3.10.2.3.15.2

Subtract $27$ from $- 1$ .

$\frac{62}{3} - \frac{- 28}{3} - 6 ((\frac{2^{2}}{2}) - \frac{{(- 1)}^{2}}{2})$

Step 3.10.2.3.16

Move the negative in front of the fraction.

$\frac{62}{3} - - \frac{28}{3} - 6 ((\frac{2^{2}}{2}) - \frac{{(- 1)}^{2}}{2})$

Step 3.10.2.3.17

Multiply $- 1$ by $- 1$ .

$\frac{62}{3} + 1 (\frac{28}{3}) - 6 ((\frac{2^{2}}{2}) - \frac{{(- 1)}^{2}}{2})$

Step 3.10.2.3.18

Multiply $\frac{28}{3}$ by $1$ .

$\frac{62}{3} + \frac{28}{3} - 6 ((\frac{2^{2}}{2}) - \frac{{(- 1)}^{2}}{2})$

Step 3.10.2.3.19

Combine the numerators over the common denominator.

$\frac{62 + 28}{3} - 6 ((\frac{2^{2}}{2}) - \frac{{(- 1)}^{2}}{2})$

Step 3.10.2.3.20

Add $62$ and $28$ .

$\frac{90}{3} - 6 ((\frac{2^{2}}{2}) - \frac{{(- 1)}^{2}}{2})$

Step 3.10.2.3.21

Cancel the common factor of $90$ and $3$ .

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Step 3.10.2.3.21.1

Factor $3$ out of $90$ .

$\frac{3 \cdot 30}{3} - 6 ((\frac{2^{2}}{2}) - \frac{{(- 1)}^{2}}{2})$

Step 3.10.2.3.21.2

Cancel the common factors.

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Step 3.10.2.3.21.2.1

Factor $3$ out of $3$ .

$\frac{3 \cdot 30}{3 (1)} - 6 ((\frac{2^{2}}{2}) - \frac{{(- 1)}^{2}}{2})$

Step 3.10.2.3.21.2.2

Cancel the common factor.

$\frac{3 \cdot 30}{3 \cdot 1} - 6 ((\frac{2^{2}}{2}) - \frac{{(- 1)}^{2}}{2})$

Step 3.10.2.3.21.2.3

Rewrite the expression.

$\frac{30}{1} - 6 ((\frac{2^{2}}{2}) - \frac{{(- 1)}^{2}}{2})$

Step 3.10.2.3.21.2.4

Divide $30$ by $1$ .

$30 - 6 ((\frac{2^{2}}{2}) - \frac{{(- 1)}^{2}}{2})$

Step 3.10.2.3.22

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

$30 - 6 (\frac{4}{2} - \frac{{(- 1)}^{2}}{2})$

Step 3.10.2.3.23

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

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Step 3.10.2.3.23.1

Factor $2$ out of $4$ .

$30 - 6 (\frac{2 \cdot 2}{2} - \frac{{(- 1)}^{2}}{2})$

Step 3.10.2.3.23.2

Cancel the common factors.

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Step 3.10.2.3.23.2.1

Factor $2$ out of $2$ .

$30 - 6 (\frac{2 \cdot 2}{2 (1)} - \frac{{(- 1)}^{2}}{2})$

Step 3.10.2.3.23.2.2

Cancel the common factor.

$30 - 6 (\frac{2 \cdot 2}{2 \cdot 1} - \frac{{(- 1)}^{2}}{2})$

Step 3.10.2.3.23.2.3

Rewrite the expression.

$30 - 6 (\frac{2}{1} - \frac{{(- 1)}^{2}}{2})$

Step 3.10.2.3.23.2.4

Divide $2$ by $1$ .

$30 - 6 (2 - \frac{{(- 1)}^{2}}{2})$

Step 3.10.2.3.24

Raise $- 1$ to the power of $2$ .

$30 - 6 (2 - \frac{1}{2})$

Step 3.10.2.3.25

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

$30 - 6 (2 \cdot \frac{2}{2} - \frac{1}{2})$

Step 3.10.2.3.26

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

$30 - 6 (\frac{2 \cdot 2}{2} - \frac{1}{2})$

Step 3.10.2.3.27

Combine the numerators over the common denominator.

$30 - 6 \frac{2 \cdot 2 - 1}{2}$

Step 3.10.2.3.28

Simplify the numerator.

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Step 3.10.2.3.28.1

Multiply $2$ by $2$ .

$30 - 6 \frac{4 - 1}{2}$

Step 3.10.2.3.28.2

Subtract $1$ from $4$ .

$30 - 6 (\frac{3}{2})$

Step 3.10.2.3.29

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

$30 + \frac{- 6 \cdot 3}{2}$

Step 3.10.2.3.30

Multiply $- 6$ by $3$ .

$30 + \frac{- 18}{2}$

Step 3.10.2.3.31

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

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Step 3.10.2.3.31.1

Factor $2$ out of $- 18$ .

$30 + \frac{2 \cdot - 9}{2}$

Step 3.10.2.3.31.2

Cancel the common factors.

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Step 3.10.2.3.31.2.1

Factor $2$ out of $2$ .

$30 + \frac{2 \cdot - 9}{2 (1)}$

Step 3.10.2.3.31.2.2

Cancel the common factor.

$30 + \frac{2 \cdot - 9}{2 \cdot 1}$

Step 3.10.2.3.31.2.3

Rewrite the expression.

$30 + \frac{- 9}{1}$

Step 3.10.2.3.31.2.4

Divide $- 9$ by $1$ .

$30 - 9$

Step 3.10.2.3.32

Subtract $9$ from $30$ .

$21$

Step 4