Calculus Examples

$y = 2 x + 24$ , $y = x^{2}$

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.

$2 x + 24 = x^{2}$

Step 1.2

Solve $2 x + 24 = x^{2}$ for $x$ .

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Step 1.2.1

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

$2 x + 24 - x^{2} = 0$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Factor $- 1$ out of $2 x + 24 - x^{2}$ .

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Step 1.2.2.1.1

Reorder the expression.

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Step 1.2.2.1.1.1

Move $24$ .

$2 x - x^{2} + 24 = 0$

Step 1.2.2.1.1.2

Reorder $2 x$ and $- x^{2}$ .

$- x^{2} + 2 x + 24 = 0$

Step 1.2.2.1.2

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) + 2 x + 24 = 0$

Step 1.2.2.1.3

Factor $- 1$ out of $2 x$ .

$- (x^{2}) - (- 2 x) + 24 = 0$

Step 1.2.2.1.4

Rewrite $24$ as $- 1 (- 24)$ .

$- (x^{2}) - (- 2 x) - 1 \cdot - 24 = 0$

Step 1.2.2.1.5

Factor $- 1$ out of $- (x^{2}) - (- 2 x)$ .

$- (x^{2} - 2 x) - 1 \cdot - 24 = 0$

Step 1.2.2.1.6

Factor $- 1$ out of $- (x^{2} - 2 x) - 1 (- 24)$ .

$- (x^{2} - 2 x - 24) = 0$

Step 1.2.2.2

Factor.

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Step 1.2.2.2.1

Factor $x^{2} - 2 x - 24$ using the AC method.

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Step 1.2.2.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 24$ and whose sum is $- 2$ .

$- 6, 4 + y = x^{2}$

Step 1.2.2.2.1.2

Write the factored form using these integers.

$- ((x - 6) (x + 4)) = 0$

Step 1.2.2.2.2

Remove unnecessary parentheses.

$- (x - 6) (x + 4) = 0$

Step 1.2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 6 = 0$

$x + 4 = 0 + y = x^{2}$

Step 1.2.4

Set $x - 6$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

Step 1.2.4.2

Add $6$ to both sides of the equation.

$x = 6$

Step 1.2.5

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 1.2.5.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 1.2.5.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 1.2.6

The final solution is all the values that make $- (x - 6) (x + 4) = 0$ true.

$x = 6, - 4$

Step 1.3

Evaluate $y$ when $x = 6$ .

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Step 1.3.1

Substitute $6$ for $x$ .

$y = {(6)}^{2}$

Step 1.3.2

Substitute $6$ for $x$ in $y = {(6)}^{2}$ and solve for $y$ .

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Step 1.3.2.1

Remove parentheses.

$y = 6^{2}$

Step 1.3.2.2

Raise $6$ to the power of $2$ .

$y = 36$

Step 1.4

Evaluate $y$ when $x = - 4$ .

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Step 1.4.1

Substitute $- 4$ for $x$ .

$y = {(- 4)}^{2}$

Step 1.4.2

Raise $- 4$ to the power of $2$ .

$y = 16$

Step 1.5

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

$(6, 36)$

$(- 4, 16)$

$(6, 36)$

$(- 4, 16)$

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.

$A r e a = \int_{- 4}^{6} 2 x + 24 d x - \int_{- 4}^{6} x^{2} d x$

Step 3

Integrate to find the area between $- 4$ and $6$ .

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Step 3.1

Combine the integrals into a single integral.

$\int_{- 4}^{6} 2 x + 24 - (x^{2}) d x$

Step 3.2

Multiply $- 1$ by $x^{2}$ .

$\int_{- 4}^{6} 2 x + 24 - x^{2} d x$

Step 3.3

Split the single integral into multiple integrals.

$\int_{- 4}^{6} 2 x d x + \int_{- 4}^{6} 24 d x + \int_{- 4}^{6} - x^{2} d x$

Step 3.4

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int_{- 4}^{6} x d x + \int_{- 4}^{6} 24 d x + \int_{- 4}^{6} - x^{2} d x$

Step 3.5

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

$2 (\frac{1}{2} x^{2}]_{- 4}^{6}) + \int_{- 4}^{6} 24 d x + \int_{- 4}^{6} - x^{2} d x$

Step 3.6

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

$2 (\frac{x^{2}}{2}]_{- 4}^{6}) + \int_{- 4}^{6} 24 d x + \int_{- 4}^{6} - x^{2} d x$

Step 3.7

Apply the constant rule.

$2 (\frac{x^{2}}{2}]_{- 4}^{6}) + 24 x]_{- 4}^{6} + \int_{- 4}^{6} - x^{2} d x$

Step 3.8

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

$2 (\frac{x^{2}}{2}]_{- 4}^{6}) + 24 x]_{- 4}^{6} - \int_{- 4}^{6} x^{2} d x$

Step 3.9

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

$2 (\frac{x^{2}}{2}]_{- 4}^{6}) + 24 x]_{- 4}^{6} - (\frac{1}{3} x^{3}]_{- 4}^{6})$

Step 3.10

Simplify the answer.

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Step 3.10.1

Combine $\frac{1}{3}$ and $x^{3}$ .

$2 (\frac{x^{2}}{2}]_{- 4}^{6}) + 24 x]_{- 4}^{6} - (\frac{x^{3}}{3}]_{- 4}^{6})$

Step 3.10.2

Substitute and simplify.

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Step 3.10.2.1

Evaluate $\frac{x^{2}}{2}$ at $6$ and at $- 4$ .

$2 ((\frac{6^{2}}{2}) - \frac{{(- 4)}^{2}}{2}) + 24 x]_{- 4}^{6} - (\frac{x^{3}}{3}]_{- 4}^{6})$

Step 3.10.2.2

Evaluate $24 x$ at $6$ and at $- 4$ .

$2 (\frac{6^{2}}{2} - \frac{{(- 4)}^{2}}{2}) + 24 \cdot 6 - 24 \cdot - 4 - (\frac{x^{3}}{3}]_{- 4}^{6})$

Step 3.10.2.3

Evaluate $\frac{x^{3}}{3}$ at $6$ and at $- 4$ .

$2 (\frac{6^{2}}{2} - \frac{{(- 4)}^{2}}{2}) + 24 \cdot 6 - 24 \cdot - 4 - ((\frac{6^{3}}{3}) - \frac{{(- 4)}^{3}}{3})$

Step 3.10.2.4

Simplify.

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Step 3.10.2.4.1

Raise $6$ to the power of $2$ .

$2 (\frac{36}{2} - \frac{{(- 4)}^{2}}{2}) + 24 \cdot 6 - 24 \cdot - 4 - ((\frac{6^{3}}{3}) - \frac{{(- 4)}^{3}}{3})$

Step 3.10.2.4.2

Cancel the common factor of $36$ and $2$ .

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Step 3.10.2.4.2.1

Factor $2$ out of $36$ .

$2 (\frac{2 \cdot 18}{2} - \frac{{(- 4)}^{2}}{2}) + 24 \cdot 6 - 24 \cdot - 4 - ((\frac{6^{3}}{3}) - \frac{{(- 4)}^{3}}{3})$

Step 3.10.2.4.2.2

Cancel the common factors.

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Step 3.10.2.4.2.2.1

Factor $2$ out of $2$ .

$2 (\frac{2 \cdot 18}{2 (1)} - \frac{{(- 4)}^{2}}{2}) + 24 \cdot 6 - 24 \cdot - 4 - ((\frac{6^{3}}{3}) - \frac{{(- 4)}^{3}}{3})$

Step 3.10.2.4.2.2.2

Cancel the common factor.

$2 (\frac{2 \cdot 18}{2 \cdot 1} - \frac{{(- 4)}^{2}}{2}) + 24 \cdot 6 - 24 \cdot - 4 - ((\frac{6^{3}}{3}) - \frac{{(- 4)}^{3}}{3})$

Step 3.10.2.4.2.2.3

Rewrite the expression.

$2 (\frac{18}{1} - \frac{{(- 4)}^{2}}{2}) + 24 \cdot 6 - 24 \cdot - 4 - ((\frac{6^{3}}{3}) - \frac{{(- 4)}^{3}}{3})$

Step 3.10.2.4.2.2.4

Divide $18$ by $1$ .

$2 (18 - \frac{{(- 4)}^{2}}{2}) + 24 \cdot 6 - 24 \cdot - 4 - ((\frac{6^{3}}{3}) - \frac{{(- 4)}^{3}}{3})$

Step 3.10.2.4.3

Raise $- 4$ to the power of $2$ .

$2 (18 - \frac{16}{2}) + 24 \cdot 6 - 24 \cdot - 4 - ((\frac{6^{3}}{3}) - \frac{{(- 4)}^{3}}{3})$

Step 3.10.2.4.4

Cancel the common factor of $16$ and $2$ .

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Step 3.10.2.4.4.1

Factor $2$ out of $16$ .

$2 (18 - \frac{2 \cdot 8}{2}) + 24 \cdot 6 - 24 \cdot - 4 - ((\frac{6^{3}}{3}) - \frac{{(- 4)}^{3}}{3})$

Step 3.10.2.4.4.2

Cancel the common factors.

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Step 3.10.2.4.4.2.1

Factor $2$ out of $2$ .

$2 (18 - \frac{2 \cdot 8}{2 (1)}) + 24 \cdot 6 - 24 \cdot - 4 - ((\frac{6^{3}}{3}) - \frac{{(- 4)}^{3}}{3})$

Step 3.10.2.4.4.2.2

Cancel the common factor.

$2 (18 - \frac{2 \cdot 8}{2 \cdot 1}) + 24 \cdot 6 - 24 \cdot - 4 - ((\frac{6^{3}}{3}) - \frac{{(- 4)}^{3}}{3})$

Step 3.10.2.4.4.2.3

Rewrite the expression.

$2 (18 - \frac{8}{1}) + 24 \cdot 6 - 24 \cdot - 4 - ((\frac{6^{3}}{3}) - \frac{{(- 4)}^{3}}{3})$

Step 3.10.2.4.4.2.4

Divide $8$ by $1$ .

$2 (18 - 1 \cdot 8) + 24 \cdot 6 - 24 \cdot - 4 - ((\frac{6^{3}}{3}) - \frac{{(- 4)}^{3}}{3})$

Step 3.10.2.4.5

Multiply $- 1$ by $8$ .

$2 (18 - 8) + 24 \cdot 6 - 24 \cdot - 4 - ((\frac{6^{3}}{3}) - \frac{{(- 4)}^{3}}{3})$

Step 3.10.2.4.6

Subtract $8$ from $18$ .

$2 \cdot 10 + 24 \cdot 6 - 24 \cdot - 4 - ((\frac{6^{3}}{3}) - \frac{{(- 4)}^{3}}{3})$

Step 3.10.2.4.7

Multiply $2$ by $10$ .

$20 + 24 \cdot 6 - 24 \cdot - 4 - ((\frac{6^{3}}{3}) - \frac{{(- 4)}^{3}}{3})$

Step 3.10.2.4.8

Multiply $24$ by $6$ .

$20 + 144 - 24 \cdot - 4 - ((\frac{6^{3}}{3}) - \frac{{(- 4)}^{3}}{3})$

Step 3.10.2.4.9

Multiply $- 24$ by $- 4$ .

$20 + 144 + 96 - ((\frac{6^{3}}{3}) - \frac{{(- 4)}^{3}}{3})$

Step 3.10.2.4.10

Add $144$ and $96$ .

$20 + 240 - ((\frac{6^{3}}{3}) - \frac{{(- 4)}^{3}}{3})$

Step 3.10.2.4.11

Add $20$ and $240$ .

$260 - ((\frac{6^{3}}{3}) - \frac{{(- 4)}^{3}}{3})$

Step 3.10.2.4.12

Raise $6$ to the power of $3$ .

$260 - (\frac{216}{3} - \frac{{(- 4)}^{3}}{3})$

Step 3.10.2.4.13

Cancel the common factor of $216$ and $3$ .

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Step 3.10.2.4.13.1

Factor $3$ out of $216$ .

$260 - (\frac{3 \cdot 72}{3} - \frac{{(- 4)}^{3}}{3})$

Step 3.10.2.4.13.2

Cancel the common factors.

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Step 3.10.2.4.13.2.1

Factor $3$ out of $3$ .

$260 - (\frac{3 \cdot 72}{3 (1)} - \frac{{(- 4)}^{3}}{3})$

Step 3.10.2.4.13.2.2

Cancel the common factor.

$260 - (\frac{3 \cdot 72}{3 \cdot 1} - \frac{{(- 4)}^{3}}{3})$

Step 3.10.2.4.13.2.3

Rewrite the expression.

$260 - (\frac{72}{1} - \frac{{(- 4)}^{3}}{3})$

Step 3.10.2.4.13.2.4

Divide $72$ by $1$ .

$260 - (72 - \frac{{(- 4)}^{3}}{3})$

Step 3.10.2.4.14

Raise $- 4$ to the power of $3$ .

$260 - (72 - \frac{- 64}{3})$

Step 3.10.2.4.15

Move the negative in front of the fraction.

$260 - (72 - - \frac{64}{3})$

Step 3.10.2.4.16

Multiply $- 1$ by $- 1$ .

$260 - (72 + 1 (\frac{64}{3}))$

Step 3.10.2.4.17

Multiply $\frac{64}{3}$ by $1$ .

$260 - (72 + \frac{64}{3})$

Step 3.10.2.4.18

To write $72$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$260 - (72 \cdot \frac{3}{3} + \frac{64}{3})$

Step 3.10.2.4.19

Combine $72$ and $\frac{3}{3}$ .

$260 - (\frac{72 \cdot 3}{3} + \frac{64}{3})$

Step 3.10.2.4.20

Combine the numerators over the common denominator.

$260 - \frac{72 \cdot 3 + 64}{3}$

Step 3.10.2.4.21

Simplify the numerator.

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Step 3.10.2.4.21.1

Multiply $72$ by $3$ .

$260 - \frac{216 + 64}{3}$

Step 3.10.2.4.21.2

Add $216$ and $64$ .

$260 - \frac{280}{3}$

Step 3.10.2.4.22

To write $260$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$260 \cdot \frac{3}{3} - \frac{280}{3}$

Step 3.10.2.4.23

Combine $260$ and $\frac{3}{3}$ .

$\frac{260 \cdot 3}{3} - \frac{280}{3}$

Step 3.10.2.4.24

Combine the numerators over the common denominator.

$\frac{260 \cdot 3 - 280}{3}$

Step 3.10.2.4.25

Simplify the numerator.

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Step 3.10.2.4.25.1

Multiply $260$ by $3$ .

$\frac{780 - 280}{3}$

Step 3.10.2.4.25.2

Subtract $280$ from $780$ .

$\frac{500}{3}$

Step 4