Find the Area Between the Curves 4x+y^2=12 , x=y

$4 x + y^{2} = 12$ , $x = y$

Solve by substitution to find the intersection between the curves.

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Substitute $y$ for $x$ into $4 x + y^{2} = 12$ then solve for $y$ .

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Replace $x$ with $y$ in the equation.

$4 (y) + y^{2} = 12$

Solve the equation for $y$ .

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Set the equation equal to zero.

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Move $12$ to the left side of the equation by subtracting it from both sides.

$4 (y) + y^{2} - 12 = 0$

Multiply $4$ by $y$ .

$4 y + y^{2} - 12 = 0$

Factor the left side of the equation.

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

$4 u + u^{2} - 12$

Factor $4 u + u^{2} - 12$ using the AC method.

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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 $- 12$ and whose sum is $4$ .

$- 2, 6$

Write the factored form using these integers.

$(u - 2) (u + 6)$

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

$(y - 2) (y + 6)$

Replace the left side with the factored expression.

$(y - 2) (y + 6) = 0$

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

$y - 2 = 0$

$y + 6 = 0$

Set the first factor equal to $0$ and solve.

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Set the first factor equal to $0$ .

$y - 2 = 0$

Add $2$ to both sides of the equation.

$y = 2$

Set the next factor equal to $0$ and solve.

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Set the next factor equal to $0$ .

$y + 6 = 0$

Subtract $6$ from both sides of the equation.

$y = - 6$

The final solution is all the values that make $(y - 2) (y + 6) = 0$ true.

$y = 2, - 6$

Substitute $2$ for $y$ into $4 x + y^{2} = 12$ then solve for $x$ .

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Replace $y$ with $2$ in the equation.

$y = 2$

$4 x + {(2)}^{2} = 12$

Solve the equation for $x$ .

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

$4 x + 4 = 12$

Move all terms not containing $x$ to the right side of the equation.

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Subtract $4$ from both sides of the equation.

$4 x = 12 - 4$

Subtract $4$ from $12$ .

$4 x = 8$

Divide each term by $4$ and simplify.

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Divide each term in $4 x = 8$ by $4$ .

$\frac{4 x}{4} = \frac{8}{4}$

Cancel the common factor of $4$ .

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Cancel the common factor.

$\frac{4 x}{4} = \frac{8}{4}$

Divide $x$ by $1$ .

$x = \frac{8}{4}$

Divide $8$ by $4$ .

$x = 2$

Substitute $- 6$ for $y$ into $4 x + y^{2} = 12$ then solve for $x$ .

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Replace $y$ with $- 6$ in the equation.

$y = - 6$

$4 x + {(- 6)}^{2} = 12$

Solve the equation for $x$ .

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

$4 x + 36 = 12$

Move all terms not containing $x$ to the right side of the equation.

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Subtract $36$ from both sides of the equation.

$4 x = 12 - 36$

Subtract $36$ from $12$ .

$4 x = - 24$

Divide each term by $4$ and simplify.

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Divide each term in $4 x = - 24$ by $4$ .

$\frac{4 x}{4} = \frac{- 24}{4}$

Cancel the common factor of $4$ .

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Cancel the common factor.

$\frac{4 x}{4} = \frac{- 24}{4}$

Divide $x$ by $1$ .

$x = \frac{- 24}{4}$

Divide $- 24$ by $4$ .

$x = - 6$

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

$(2, 2)$

$(- 6, - 6)$

$(2, 2)$

$(- 6, - 6)$

Solve $4 x + y^{2} = 12$ in terms of $x$ .

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

$4 x = 12 - y^{2}$

Divide each term by $4$ and simplify.

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Divide each term in $4 x = 12 - y^{2}$ by $4$ .

$\frac{4 x}{4} = \frac{12}{4} + \frac{- y^{2}}{4}$

Cancel the common factor of $4$ .

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Cancel the common factor.

$\frac{4 x}{4} = \frac{12}{4} + \frac{- y^{2}}{4}$

Divide $x$ by $1$ .

$x = \frac{12}{4} + \frac{- y^{2}}{4}$

Simplify each term.

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Divide $12$ by $4$ .

$x = 3 + \frac{- y^{2}}{4}$

Move the negative in front of the fraction.

$x = 3 - \frac{y^{2}}{4}$

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.

$A r e a = \int_{- 6}^{2} 3 - \frac{y^{2}}{4} d y - \int_{- 6}^{2} y d y$

Integrate to find the area between $- 6$ and $2$ .

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Combine the integrals into a single integral.

$\int_{- 6}^{2} 3 - \frac{y^{2}}{4} - (y) d y$

Multiply $- 1$ by $y$ .

$\int_{- 6}^{2} 3 - \frac{y^{2}}{4} - y d y$

Split the single integral into multiple integrals.

$\int_{- 6}^{2} 3 d y + \int_{- 6}^{2} - \frac{y^{2}}{4} d y + \int_{- 6}^{2} - y d y$

Since $3$ is constant with respect to $y$ , move $3$ out of the integral.

$3 y]_{- 6}^{2} + \int_{- 6}^{2} - \frac{y^{2}}{4} d y + \int_{- 6}^{2} - y d y$

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

$3 y]_{- 6}^{2} - \int_{- 6}^{2} \frac{y^{2}}{4} d y + \int_{- 6}^{2} - y d y$

Since $\frac{1}{4}$ is constant with respect to $y$ , move $\frac{1}{4}$ out of the integral.

$3 y]_{- 6}^{2} - (\frac{1}{4} \int_{- 6}^{2} y^{2} d y) + \int_{- 6}^{2} - y d y$

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

$3 y]_{- 6}^{2} - \frac{1}{4} \frac{1}{3} y^{3}]_{- 6}^{2} + \int_{- 6}^{2} - y d y$

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

$3 y]_{- 6}^{2} - \frac{1}{4} \frac{1}{3} y^{3}]_{- 6}^{2} - \int_{- 6}^{2} y d y$

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

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

Simplify the answer.

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

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

Substitute and simplify.

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Evaluate $3 y$ at $2$ and at $- 6$ .

$(3 \cdot 2) - 3 \cdot - 6 - \frac{1}{4} \frac{1}{3} y^{3}]_{- 6}^{2} - (\frac{y^{2}}{2}]_{- 6}^{2})$

Evaluate $\frac{1}{3} y^{3}$ at $2$ and at $- 6$ .

$3 \cdot 2 - 3 \cdot - 6 - \frac{1}{4} (\frac{1}{3} \cdot 2^{3} - \frac{1}{3} {(- 6)}^{3}) - (\frac{y^{2}}{2}]_{- 6}^{2})$

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

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

Simplify.

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Multiply $3$ by $2$ .

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

Multiply $- 3$ by $- 6$ .

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

Add $6$ and $18$ .

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

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

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

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

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

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

$24 - \frac{1}{4} (\frac{8}{3} - \frac{1}{3} \cdot - 216) - ((\frac{2^{2}}{2}) - \frac{{(- 6)}^{2}}{2})$

Multiply $- 216$ by $- 1$ .

$24 - \frac{1}{4} (\frac{8}{3} + 216 (\frac{1}{3})) - ((\frac{2^{2}}{2}) - \frac{{(- 6)}^{2}}{2})$

Combine $216$ and $\frac{1}{3}$ .

$24 - \frac{1}{4} (\frac{8}{3} + \frac{216}{3}) - ((\frac{2^{2}}{2}) - \frac{{(- 6)}^{2}}{2})$

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

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Factor $3$ out of $216$ .

$24 - \frac{1}{4} (\frac{8}{3} + \frac{3 \cdot 72}{3}) - ((\frac{2^{2}}{2}) - \frac{{(- 6)}^{2}}{2})$

Cancel the common factors.

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Factor $3$ out of $3$ .

$24 - \frac{1}{4} (\frac{8}{3} + \frac{3 \cdot 72}{3 (1)}) - ((\frac{2^{2}}{2}) - \frac{{(- 6)}^{2}}{2})$

Cancel the common factor.

$24 - \frac{1}{4} (\frac{8}{3} + \frac{3 \cdot 72}{3 \cdot 1}) - ((\frac{2^{2}}{2}) - \frac{{(- 6)}^{2}}{2})$

Rewrite the expression.

$24 - \frac{1}{4} (\frac{8}{3} + \frac{72}{1}) - ((\frac{2^{2}}{2}) - \frac{{(- 6)}^{2}}{2})$

Divide $72$ by $1$ .

$24 - \frac{1}{4} (\frac{8}{3} + 72) - ((\frac{2^{2}}{2}) - \frac{{(- 6)}^{2}}{2})$

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

$24 - \frac{1}{4} (\frac{8}{3} + 72 \cdot \frac{3}{3}) - ((\frac{2^{2}}{2}) - \frac{{(- 6)}^{2}}{2})$

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

$24 - \frac{1}{4} (\frac{8}{3} + \frac{72 \cdot 3}{3}) - ((\frac{2^{2}}{2}) - \frac{{(- 6)}^{2}}{2})$

Combine the numerators over the common denominator.

$24 - \frac{1}{4} \cdot \frac{8 + 72 \cdot 3}{3} - ((\frac{2^{2}}{2}) - \frac{{(- 6)}^{2}}{2})$

Simplify the numerator.

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Multiply $72$ by $3$ .

$24 - \frac{1}{4} \cdot \frac{8 + 216}{3} - ((\frac{2^{2}}{2}) - \frac{{(- 6)}^{2}}{2})$

Add $8$ and $216$ .

$24 - \frac{1}{4} \cdot \frac{224}{3} - ((\frac{2^{2}}{2}) - \frac{{(- 6)}^{2}}{2})$

Multiply $\frac{224}{3}$ and $\frac{1}{4}$ .

$24 - \frac{224}{3 \cdot 4} - ((\frac{2^{2}}{2}) - \frac{{(- 6)}^{2}}{2})$

Multiply $3$ by $4$ .

$24 - \frac{224}{12} - ((\frac{2^{2}}{2}) - \frac{{(- 6)}^{2}}{2})$

Cancel the common factor of $224$ and $12$ .

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Factor $4$ out of $224$ .

$24 - \frac{4 (56)}{12} - ((\frac{2^{2}}{2}) - \frac{{(- 6)}^{2}}{2})$

Cancel the common factors.

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Factor $4$ out of $12$ .

$24 - \frac{4 \cdot 56}{4 \cdot 3} - ((\frac{2^{2}}{2}) - \frac{{(- 6)}^{2}}{2})$

Cancel the common factor.

$24 - \frac{4 \cdot 56}{4 \cdot 3} - ((\frac{2^{2}}{2}) - \frac{{(- 6)}^{2}}{2})$

Rewrite the expression.

$24 - \frac{56}{3} - ((\frac{2^{2}}{2}) - \frac{{(- 6)}^{2}}{2})$

To write $24$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$24 \cdot \frac{3}{3} - \frac{56}{3} - ((\frac{2^{2}}{2}) - \frac{{(- 6)}^{2}}{2})$

Combine $24$ and $\frac{3}{3}$ .

$\frac{24 \cdot 3}{3} - \frac{56}{3} - ((\frac{2^{2}}{2}) - \frac{{(- 6)}^{2}}{2})$

Combine the numerators over the common denominator.

$\frac{24 \cdot 3 - 56}{3} - ((\frac{2^{2}}{2}) - \frac{{(- 6)}^{2}}{2})$

Simplify the numerator.

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Multiply $24$ by $3$ .

$\frac{72 - 56}{3} - ((\frac{2^{2}}{2}) - \frac{{(- 6)}^{2}}{2})$

Subtract $56$ from $72$ .

$\frac{16}{3} - ((\frac{2^{2}}{2}) - \frac{{(- 6)}^{2}}{2})$

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

$\frac{16}{3} - (\frac{4}{2} - \frac{{(- 6)}^{2}}{2})$

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

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Factor $2$ out of $4$ .

$\frac{16}{3} - (\frac{2 \cdot 2}{2} - \frac{{(- 6)}^{2}}{2})$

Cancel the common factors.

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Factor $2$ out of $2$ .

$\frac{16}{3} - (\frac{2 \cdot 2}{2 (1)} - \frac{{(- 6)}^{2}}{2})$

Cancel the common factor.

$\frac{16}{3} - (\frac{2 \cdot 2}{2 \cdot 1} - \frac{{(- 6)}^{2}}{2})$

Rewrite the expression.

$\frac{16}{3} - (\frac{2}{1} - \frac{{(- 6)}^{2}}{2})$

Divide $2$ by $1$ .

$\frac{16}{3} - (2 - \frac{{(- 6)}^{2}}{2})$

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

$\frac{16}{3} - (2 - \frac{36}{2})$

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

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Factor $2$ out of $36$ .

$\frac{16}{3} - (2 - \frac{2 \cdot 18}{2})$

Cancel the common factors.

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Factor $2$ out of $2$ .

$\frac{16}{3} - (2 - \frac{2 \cdot 18}{2 (1)})$

Cancel the common factor.

$\frac{16}{3} - (2 - \frac{2 \cdot 18}{2 \cdot 1})$

Rewrite the expression.

$\frac{16}{3} - (2 - \frac{18}{1})$

Divide $18$ by $1$ .

$\frac{16}{3} - (2 - 1 \cdot 18)$

Multiply $- 1$ by $18$ .

$\frac{16}{3} - (2 - 18)$

Subtract $18$ from $2$ .

$\frac{16}{3} - - 16$

Multiply $- 1$ by $- 16$ .

$\frac{16}{3} + 16$

To write $16$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{16}{3} + 16 \cdot \frac{3}{3}$

Combine $16$ and $\frac{3}{3}$ .

$\frac{16}{3} + \frac{16 \cdot 3}{3}$

Combine the numerators over the common denominator.

$\frac{16 + 16 \cdot 3}{3}$

Simplify the numerator.

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Multiply $16$ by $3$ .

$\frac{16 + 48}{3}$

Add $16$ and $48$ .

$\frac{64}{3}$

Find the Area Between the Curves 4x+y^2=12 , x=y

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