Calculus Examples

$2 x + y^{2} = 8$ , $x = y$

Step 1

Solve by substitution to find the intersection between the curves.

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Step 1.1

Replace all occurrences of $x$ with $y$ in each equation.

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Step 1.1.1

Replace all occurrences of $x$ in $2 x + y^{2} = 8$ with $y$ .

$2 (y) + y^{2} = 8$

$x = y$

Step 1.1.2

Simplify the left side.

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Step 1.1.2.1

Multiply $2$ by $y$ .

$2 y + y^{2} = 8$

$x = y$

$2 y + y^{2} = 8$

$x = y$

$2 y + y^{2} = 8$

$x = y$

Step 1.2

Solve for $y$ in $2 y + y^{2} = 8$ .

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Step 1.2.1

Subtract $8$ from both sides of the equation.

$2 y + y^{2} - 8 = 0$

$x = y$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Let $u = y$ . Substitute $u$ for all occurrences of $y$ .

$2 u + u^{2} - 8 = 0$

$x = y$

Step 1.2.2.2

Factor $2 u + u^{2} - 8$ using the AC method.

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Step 1.2.2.2.1

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 $- 8$ and whose sum is $2$ .

$- 2, 4$

$x = y$

Step 1.2.2.2.2

Write the factored form using these integers.

$(u - 2) (u + 4) = 0$

$x = y$

$(u - 2) (u + 4) = 0$

$x = y$

Step 1.2.2.3

Replace all occurrences of $u$ with $y$ .

$(y - 2) (y + 4) = 0$

$x = y$

$(y - 2) (y + 4) = 0$

$x = y$

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 $0$ .

$y - 2 = 0$

$y + 4 = 0$

$x = y$

Step 1.2.4

Set $y - 2$ equal to $0$ and solve for $y$ .

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Step 1.2.4.1

Set $y - 2$ equal to $0$ .

$y - 2 = 0$

$x = y$

Step 1.2.4.2

Add $2$ to both sides of the equation.

$y = 2$

$x = y$

$y = 2$

$x = y$

Step 1.2.5

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

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Step 1.2.5.1

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

$y + 4 = 0$

$x = y$

Step 1.2.5.2

Subtract $4$ from both sides of the equation.

$y = - 4$

$x = y$

$y = - 4$

$x = y$

Step 1.2.6

The final solution is all the values that make $(y - 2) (y + 4) = 0$ true.

$y = 2, - 4$

$x = y$

$y = 2, - 4$

$x = y$

Step 1.3

Replace all occurrences of $y$ with $2$ in each equation.

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Step 1.3.1

Replace all occurrences of $y$ in $x = y$ with $2$ .

$x = 2$

$y = 2$

Step 1.3.2

Simplify the left side.

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Step 1.3.2.1

Remove parentheses.

$x = 2$

$y = 2$

$x = 2$

$y = 2$

$x = 2$

$y = 2$

Step 1.4

Replace all occurrences of $y$ with $- 4$ in each equation.

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Step 1.4.1

Replace all occurrences of $y$ in $x = y$ with $- 4$ .

$x = - 4$

$y = - 4$

Step 1.4.2

Simplify the left side.

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Step 1.4.2.1

Remove parentheses.

$x = - 4$

$y = - 4$

$x = - 4$

$y = - 4$

$x = - 4$

$y = - 4$

Step 1.5

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

$(2, 2)$

$(- 4, - 4)$

$(2, 2)$

$(- 4, - 4)$

Step 2

Solve $2 x + y^{2} = 8$ in terms of $x$ .

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Step 2.1

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

$2 x = 8 - y^{2}$

Step 2.2

Divide each term in $2 x = 8 - y^{2}$ by $2$ and simplify.

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Step 2.2.1

Divide each term in $2 x = 8 - y^{2}$ by $2$ .

$\frac{2 x}{2} = \frac{8}{2} + \frac{- y^{2}}{2}$

Step 2.2.2

Simplify the left side.

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Step 2.2.2.1

Cancel the common factor of $2$ .

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Step 2.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{8}{2} + \frac{- y^{2}}{2}$

Step 2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{8}{2} + \frac{- y^{2}}{2}$

Step 2.2.3

Simplify the right side.

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Divide $8$ by $2$ .

$x = 4 + \frac{- y^{2}}{2}$

Step 2.2.3.1.2

Move the negative in front of the fraction.

$x = 4 - \frac{y^{2}}{2}$

Step 3

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.

$A r e a = \int_{- 4}^{2} 4 - \frac{y^{2}}{2} d y - \int_{- 4}^{2} y d y$

Step 4

Integrate to find the area between $- 4$ and $2$ .

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Step 4.1

Combine the integrals into a single integral.

$\int_{- 4}^{2} 4 - \frac{y^{2}}{2} - (y) d y$

Step 4.2

Multiply $- 1$ by $y$ .

$\int_{- 4}^{2} 4 - \frac{y^{2}}{2} - y d y$

Step 4.3

Split the single integral into multiple integrals.

$\int_{- 4}^{2} 4 d y + \int_{- 4}^{2} - \frac{y^{2}}{2} d y + \int_{- 4}^{2} - y d y$

Step 4.4

Apply the constant rule.

$4 y]_{- 4}^{2} + \int_{- 4}^{2} - \frac{y^{2}}{2} d y + \int_{- 4}^{2} - y d y$

Step 4.5

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

$4 y]_{- 4}^{2} - \int_{- 4}^{2} \frac{y^{2}}{2} d y + \int_{- 4}^{2} - y d y$

Step 4.6

Since $\frac{1}{2}$ is constant with respect to $y$ , move $\frac{1}{2}$ out of the integral.

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

Step 4.7

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

$4 y]_{- 4}^{2} - \frac{1}{2} \frac{1}{3} y^{3}]_{- 4}^{2} + \int_{- 4}^{2} - y d y$

Step 4.8

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

$4 y]_{- 4}^{2} - \frac{1}{2} \frac{1}{3} y^{3}]_{- 4}^{2} - \int_{- 4}^{2} y d y$

Step 4.9

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

$4 y]_{- 4}^{2} - \frac{1}{2} \frac{1}{3} y^{3}]_{- 4}^{2} - (\frac{1}{2} y^{2}]_{- 4}^{2})$

Step 4.10

Simplify the answer.

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Step 4.10.1

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

$4 y]_{- 4}^{2} - \frac{1}{2} \frac{1}{3} y^{3}]_{- 4}^{2} - (\frac{y^{2}}{2}]_{- 4}^{2})$

Step 4.10.2

Substitute and simplify.

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Step 4.10.2.1

Evaluate $4 y$ at $2$ and at $- 4$ .

$(4 \cdot 2) - 4 \cdot - 4 - \frac{1}{2} \frac{1}{3} y^{3}]_{- 4}^{2} - (\frac{y^{2}}{2}]_{- 4}^{2})$

Step 4.10.2.2

Evaluate $\frac{1}{3} y^{3}$ at $2$ and at $- 4$ .

$4 \cdot 2 - 4 \cdot - 4 - \frac{1}{2} (\frac{1}{3} \cdot 2^{3} - \frac{1}{3} {(- 4)}^{3}) - (\frac{y^{2}}{2}]_{- 4}^{2})$

Step 4.10.2.3

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

$4 \cdot 2 - 4 \cdot - 4 - \frac{1}{2} (\frac{1}{3} \cdot 2^{3} - \frac{1}{3} {(- 4)}^{3}) - ((\frac{2^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 4.10.2.4

Simplify.

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Step 4.10.2.4.1

Multiply $4$ by $2$ .

$8 - 4 \cdot - 4 - \frac{1}{2} (\frac{1}{3} \cdot 2^{3} - \frac{1}{3} {(- 4)}^{3}) - ((\frac{2^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 4.10.2.4.2

Multiply $- 4$ by $- 4$ .

$8 + 16 - \frac{1}{2} (\frac{1}{3} \cdot 2^{3} - \frac{1}{3} {(- 4)}^{3}) - ((\frac{2^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 4.10.2.4.3

Add $8$ and $16$ .

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

Step 4.10.2.4.4

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

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

Step 4.10.2.4.5

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

$24 - \frac{1}{2} (\frac{8}{3} - \frac{1}{3} {(- 4)}^{3}) - ((\frac{2^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 4.10.2.4.6

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

$24 - \frac{1}{2} (\frac{8}{3} - \frac{1}{3} \cdot - 64) - ((\frac{2^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 4.10.2.4.7

Multiply $- 64$ by $- 1$ .

$24 - \frac{1}{2} (\frac{8}{3} + 64 (\frac{1}{3})) - ((\frac{2^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 4.10.2.4.8

Combine $64$ and $\frac{1}{3}$ .

$24 - \frac{1}{2} (\frac{8}{3} + \frac{64}{3}) - ((\frac{2^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 4.10.2.4.9

Combine the numerators over the common denominator.

$24 - \frac{1}{2} \cdot \frac{8 + 64}{3} - ((\frac{2^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 4.10.2.4.10

Add $8$ and $64$ .

$24 - \frac{1}{2} \cdot \frac{72}{3} - ((\frac{2^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 4.10.2.4.11

Cancel the common factor of $72$ and $3$ .

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Step 4.10.2.4.11.1

Factor $3$ out of $72$ .

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

Step 4.10.2.4.11.2

Cancel the common factors.

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Step 4.10.2.4.11.2.1

Factor $3$ out of $3$ .

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

Step 4.10.2.4.11.2.2

Cancel the common factor.

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

Step 4.10.2.4.11.2.3

Rewrite the expression.

$24 - \frac{1}{2} \cdot \frac{24}{1} - ((\frac{2^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 4.10.2.4.11.2.4

Divide $24$ by $1$ .

$24 - \frac{1}{2} \cdot 24 - ((\frac{2^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 4.10.2.4.12

Multiply $24$ by $- 1$ .

$24 - 24 (\frac{1}{2}) - ((\frac{2^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 4.10.2.4.13

Combine $- 24$ and $\frac{1}{2}$ .

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

Step 4.10.2.4.14

Cancel the common factor of $- 24$ and $2$ .

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Step 4.10.2.4.14.1

Factor $2$ out of $- 24$ .

$24 + \frac{2 \cdot - 12}{2} - ((\frac{2^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 4.10.2.4.14.2

Cancel the common factors.

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Step 4.10.2.4.14.2.1

Factor $2$ out of $2$ .

$24 + \frac{2 \cdot - 12}{2 (1)} - ((\frac{2^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 4.10.2.4.14.2.2

Cancel the common factor.

$24 + \frac{2 \cdot - 12}{2 \cdot 1} - ((\frac{2^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 4.10.2.4.14.2.3

Rewrite the expression.

$24 + \frac{- 12}{1} - ((\frac{2^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 4.10.2.4.14.2.4

Divide $- 12$ by $1$ .

$24 - 12 - ((\frac{2^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 4.10.2.4.15

Subtract $12$ from $24$ .

$12 - ((\frac{2^{2}}{2}) - \frac{{(- 4)}^{2}}{2})$

Step 4.10.2.4.16

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

$12 - (\frac{4}{2} - \frac{{(- 4)}^{2}}{2})$

Step 4.10.2.4.17

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

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Step 4.10.2.4.17.1

Factor $2$ out of $4$ .

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

Step 4.10.2.4.17.2

Cancel the common factors.

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Step 4.10.2.4.17.2.1

Factor $2$ out of $2$ .

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

Step 4.10.2.4.17.2.2

Cancel the common factor.

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

Step 4.10.2.4.17.2.3

Rewrite the expression.

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

Step 4.10.2.4.17.2.4

Divide $2$ by $1$ .

$12 - (2 - \frac{{(- 4)}^{2}}{2})$

Step 4.10.2.4.18

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

$12 - (2 - \frac{16}{2})$

Step 4.10.2.4.19

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

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Step 4.10.2.4.19.1

Factor $2$ out of $16$ .

$12 - (2 - \frac{2 \cdot 8}{2})$

Step 4.10.2.4.19.2

Cancel the common factors.

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Step 4.10.2.4.19.2.1

Factor $2$ out of $2$ .

$12 - (2 - \frac{2 \cdot 8}{2 (1)})$

Step 4.10.2.4.19.2.2

Cancel the common factor.

$12 - (2 - \frac{2 \cdot 8}{2 \cdot 1})$

Step 4.10.2.4.19.2.3

Rewrite the expression.

$12 - (2 - \frac{8}{1})$

Step 4.10.2.4.19.2.4

Divide $8$ by $1$ .

$12 - (2 - 1 \cdot 8)$

Step 4.10.2.4.20

Multiply $- 1$ by $8$ .

$12 - (2 - 8)$

Step 4.10.2.4.21

Subtract $8$ from $2$ .

$12 - - 6$

Step 4.10.2.4.22

Multiply $- 1$ by $- 6$ .

$12 + 6$

Step 4.10.2.4.23

Add $12$ and $6$ .

$18$

Step 5