Calculus Examples

$y = \frac{4}{9} x^{2}$ , $y = \frac{13}{9} - x^{2}$

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.

$\frac{4}{9} x^{2} = \frac{13}{9} - x^{2}$

Step 1.2

Solve $\frac{4}{9} x^{2} = \frac{13}{9} - x^{2}$ for $x$ .

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Step 1.2.1

Simplify $\frac{4}{9} x^{2}$ .

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Step 1.2.1.1

Rewrite.

$0 + 0 + \frac{4}{9} x^{2} = \frac{13}{9} - x^{2}$

Step 1.2.1.2

Simplify by adding zeros.

$\frac{4}{9} x^{2} = \frac{13}{9} - x^{2}$

Step 1.2.1.3

Combine $\frac{4}{9}$ and $x^{2}$ .

$\frac{4 x^{2}}{9} = \frac{13}{9} - x^{2}$

Step 1.2.2

Move all terms containing $x$ to the left side of the equation.

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Step 1.2.2.1

Add $x^{2}$ to both sides of the equation.

$\frac{4 x^{2}}{9} + x^{2} = \frac{13}{9}$

Step 1.2.2.2

To write $x^{2}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{4 x^{2}}{9} + x^{2} \cdot \frac{9}{9} = \frac{13}{9}$

Step 1.2.2.3

Combine $x^{2}$ and $\frac{9}{9}$ .

$\frac{4 x^{2}}{9} + \frac{x^{2} \cdot 9}{9} = \frac{13}{9}$

Step 1.2.2.4

Combine the numerators over the common denominator.

$\frac{4 x^{2} + x^{2} \cdot 9}{9} = \frac{13}{9}$

Step 1.2.2.5

Simplify the numerator.

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Step 1.2.2.5.1

Move $9$ to the left of $x^{2}$ .

$\frac{4 x^{2} + 9 \cdot x^{2}}{9} = \frac{13}{9}$

Step 1.2.2.5.2

Add $4 x^{2}$ and $9 x^{2}$ .

$\frac{13 x^{2}}{9} = \frac{13}{9}$

Step 1.2.3

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$13 x^{2} = 13$

Step 1.2.4

Divide each term in $13 x^{2} = 13$ by $13$ and simplify.

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Step 1.2.4.1

Divide each term in $13 x^{2} = 13$ by $13$ .

$\frac{13 x^{2}}{13} = \frac{13}{13}$

Step 1.2.4.2

Simplify the left side.

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Step 1.2.4.2.1

Cancel the common factor of $13$ .

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Step 1.2.4.2.1.1

Cancel the common factor.

$\frac{13 x^{2}}{13} = \frac{13}{13}$

Step 1.2.4.2.1.2

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{13}{13}$

Step 1.2.4.3

Simplify the right side.

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Step 1.2.4.3.1

Divide $13$ by $13$ .

$x^{2} = 1$

Step 1.2.5

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{1}$

Step 1.2.6

Any root of $1$ is $1$ .

$x = \pm 1$

Step 1.2.7

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.7.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 1$

Step 1.2.7.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 1$

Step 1.2.7.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 1, - 1$

Step 1.3

Evaluate $y$ when $x = 1$ .

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Step 1.3.1

Substitute $1$ for $x$ .

$y = \frac{13}{9} - {(1)}^{2}$

Step 1.3.2

Substitute $1$ for $x$ in $y = \frac{13}{9} - {(1)}^{2}$ and solve for $y$ .

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Step 1.3.2.1

Remove parentheses.

$y = \frac{13}{9} - 1^{2}$

Step 1.3.2.2

Simplify $\frac{13}{9} - 1^{2}$ .

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Step 1.3.2.2.1

Simplify each term.

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Step 1.3.2.2.1.1

One to any power is one.

$y = \frac{13}{9} - 1 \cdot 1$

Step 1.3.2.2.1.2

Multiply $- 1$ by $1$ .

$y = \frac{13}{9} - 1$

Step 1.3.2.2.2

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$y = \frac{13}{9} - 1 \cdot \frac{9}{9}$

Step 1.3.2.2.3

Combine $- 1$ and $\frac{9}{9}$ .

$y = \frac{13}{9} + \frac{- 1 \cdot 9}{9}$

Step 1.3.2.2.4

Combine the numerators over the common denominator.

$y = \frac{13 - 1 \cdot 9}{9}$

Step 1.3.2.2.5

Simplify the numerator.

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Step 1.3.2.2.5.1

Multiply $- 1$ by $9$ .

$y = \frac{13 - 9}{9}$

Step 1.3.2.2.5.2

Subtract $9$ from $13$ .

$y = \frac{4}{9}$

Step 1.4

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

$(1, \frac{4}{9})$

$(- 1, \frac{4}{9})$

$(1, \frac{4}{9})$

$(- 1, \frac{4}{9})$

Step 2

Combine $\frac{4}{9}$ and $x^{2}$ .

$y = \frac{4 x^{2}}{9}$

Step 3

Reorder $\frac{13}{9}$ and $- x^{2}$ .

$y = - x^{2} + \frac{13}{9}$

Step 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.

$A r e a = \int_{- 1}^{1} - x^{2} + \frac{13}{9} d x - \int_{- 1}^{1} \frac{4 x^{2}}{9} d x$

Step 5

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

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Step 5.1

Combine the integrals into a single integral.

$\int_{- 1}^{1} - x^{2} + \frac{13}{9} - (\frac{4 x^{2}}{9}) d x$

Step 5.2

To write $- x^{2}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\int_{- 1}^{1} - x^{2} \cdot \frac{9}{9} - \frac{4 x^{2}}{9} + \frac{13}{9} d x$

Step 5.3

Simplify terms.

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Step 5.3.1

Combine $- x^{2}$ and $\frac{9}{9}$ .

$\frac{- x^{2} \cdot 9}{9} - \frac{4 x^{2}}{9} + \frac{13}{9}$

Step 5.3.2

Combine the numerators over the common denominator.

$\frac{- x^{2} \cdot 9 - 4 x^{2}}{9} + \frac{13}{9}$

Step 5.3.3

Combine the numerators over the common denominator.

$\frac{- x^{2} \cdot 9 - 4 x^{2} + 13}{9}$

Step 5.3.4

Multiply $9$ by $- 1$ .

$\frac{- 9 x^{2} - 4 x^{2} + 13}{9}$

Step 5.3.5

Subtract $4 x^{2}$ from $- 9 x^{2}$ .

$\frac{- 13 x^{2} + 13}{9}$

$\int_{- 1}^{1} \frac{- 13 x^{2} + 13}{9} d x$

Step 5.4

Simplify the numerator.

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Step 5.4.1

Factor $13$ out of $- 13 x^{2} + 13$ .

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Step 5.4.1.1

Factor $13$ out of $- 13 x^{2}$ .

$\frac{13 (- x^{2}) + 13}{9}$

Step 5.4.1.2

Factor $13$ out of $13$ .

$\frac{13 (- x^{2}) + 13 (1)}{9}$

Step 5.4.1.3

Factor $13$ out of $13 (- x^{2}) + 13 (1)$ .

$\frac{13 (- x^{2} + 1)}{9}$

Step 5.4.2

Rewrite $1$ as $1^{2}$ .

$\frac{13 (- x^{2} + 1^{2})}{9}$

Step 5.4.3

Reorder $- x^{2}$ and $1^{2}$ .

$\frac{13 (1^{2} - x^{2})}{9}$

Step 5.4.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = x$ .

$\frac{13 (1 + x) (1 - x)}{9}$

$\int_{- 1}^{1} \frac{13 (1 + x) (1 - x)}{9} d x$

Step 5.5

Since $\frac{13}{9}$ is constant with respect to $x$ , move $\frac{13}{9}$ out of the integral.

$\frac{13}{9} \int_{- 1}^{1} (1 + x) (1 - x) d x$

Step 5.6

Simplify.

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Step 5.6.1

Apply the distributive property.

$\frac{13}{9} \int_{- 1}^{1} 1 (1 - x) + x (1 - x) d x$

Step 5.6.2

Apply the distributive property.

$\frac{13}{9} \int_{- 1}^{1} 1 \cdot 1 + 1 (- x) + x (1 - x) d x$

Step 5.6.3

Apply the distributive property.

$\frac{13}{9} \int_{- 1}^{1} 1 \cdot 1 + 1 (- x) + x \cdot 1 + x (- x) d x$

Step 5.6.4

Reorder $x$ and $1$ .

$\frac{13}{9} \int_{- 1}^{1} 1 \cdot 1 + 1 \cdot - 1 x + 1 \cdot x + x (- x) d x$

Step 5.6.5

Reorder $x$ and $- 1$ .

$\frac{13}{9} \int_{- 1}^{1} 1 \cdot 1 + 1 \cdot - 1 x + 1 \cdot x - 1 \cdot x \cdot x d x$

Step 5.6.6

Multiply $1$ by $1$ .

$\frac{13}{9} \int_{- 1}^{1} 1 + 1 \cdot - 1 x + 1 x - 1 x \cdot x d x$

Step 5.6.7

Multiply $- 1$ by $1$ .

$\frac{13}{9} \int_{- 1}^{1} 1 - x + 1 x - 1 x \cdot x d x$

Step 5.6.8

Multiply $x$ by $1$ .

$\frac{13}{9} \int_{- 1}^{1} 1 - x + x - 1 x \cdot x d x$

Step 5.6.9

Factor out negative.

$\frac{13}{9} \int_{- 1}^{1} 1 - x + x - (x \cdot x) d x$

Step 5.6.10

Raise $x$ to the power of $1$ .

$\frac{13}{9} \int_{- 1}^{1} 1 - x + x - (x^{1} x) d x$

Step 5.6.11

Raise $x$ to the power of $1$ .

$\frac{13}{9} \int_{- 1}^{1} 1 - x + x - (x^{1} x^{1}) d x$

Step 5.6.12

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{13}{9} \int_{- 1}^{1} 1 - x + x - x^{1 + 1} d x$

Step 5.6.13

Add $1$ and $1$ .

$\frac{13}{9} \int_{- 1}^{1} 1 - x + x - x^{2} d x$

Step 5.6.14

Add $- x$ and $x$ .

$\frac{13}{9} \int_{- 1}^{1} 1 + 0 - x^{2} d x$

Step 5.6.15

Subtract $x^{2}$ from $0$ .

$\frac{13}{9} \int_{- 1}^{1} 1 - x^{2} d x$

Step 5.6.16

Reorder $1$ and $- x^{2}$ .

$\frac{13}{9} \int_{- 1}^{1} - x^{2} + 1 d x$

Step 5.7

Split the single integral into multiple integrals.

$\frac{13}{9} (\int_{- 1}^{1} - x^{2} d x + \int_{- 1}^{1} d x)$

Step 5.8

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

$\frac{13}{9} (- \int_{- 1}^{1} x^{2} d x + \int_{- 1}^{1} d x)$

Step 5.9

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

$\frac{13}{9} (- (\frac{1}{3} x^{3}]_{- 1}^{1}) + \int_{- 1}^{1} d x)$

Step 5.10

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

$\frac{13}{9} (- (\frac{x^{3}}{3}]_{- 1}^{1}) + \int_{- 1}^{1} d x)$

Step 5.11

Apply the constant rule.

$\frac{13}{9} (- (\frac{x^{3}}{3}]_{- 1}^{1}) + x]_{- 1}^{1})$

Step 5.12

Substitute and simplify.

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Step 5.12.1

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

$\frac{13}{9} (- ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3}) + x]_{- 1}^{1})$

Step 5.12.2

Evaluate $x$ at $1$ and at $- 1$ .

$\frac{13}{9} (- (\frac{1^{3}}{3} - \frac{{(- 1)}^{3}}{3}) + 1 + 1)$

Step 5.12.3

Simplify.

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Step 5.12.3.1

One to any power is one.

$\frac{13}{9} (- (\frac{1}{3} - \frac{{(- 1)}^{3}}{3}) + 1 + 1)$

Step 5.12.3.2

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

$\frac{13}{9} (- (\frac{1}{3} - \frac{- 1}{3}) + 1 + 1)$

Step 5.12.3.3

Move the negative in front of the fraction.

$\frac{13}{9} (- (\frac{1}{3} - - \frac{1}{3}) + 1 + 1)$

Step 5.12.3.4

Multiply $- 1$ by $- 1$ .

$\frac{13}{9} (- (\frac{1}{3} + 1 (\frac{1}{3})) + 1 + 1)$

Step 5.12.3.5

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

$\frac{13}{9} (- (\frac{1}{3} + \frac{1}{3}) + 1 + 1)$

Step 5.12.3.6

Combine the numerators over the common denominator.

$\frac{13}{9} (- \frac{1 + 1}{3} + 1 + 1)$

Step 5.12.3.7

Add $1$ and $1$ .

$\frac{13}{9} (- \frac{2}{3} + 1 + 1)$

Step 5.12.3.8

Add $1$ and $1$ .

$\frac{13}{9} (- \frac{2}{3} + 2)$

Step 5.12.3.9

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{13}{9} (- \frac{2}{3} + 2 \cdot \frac{3}{3})$

Step 5.12.3.10

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

$\frac{13}{9} (- \frac{2}{3} + \frac{2 \cdot 3}{3})$

Step 5.12.3.11

Combine the numerators over the common denominator.

$\frac{13}{9} \cdot \frac{- 2 + 2 \cdot 3}{3}$

Step 5.12.3.12

Simplify the numerator.

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Step 5.12.3.12.1

Multiply $2$ by $3$ .

$\frac{13}{9} \cdot \frac{- 2 + 6}{3}$

Step 5.12.3.12.2

Add $- 2$ and $6$ .

$\frac{13}{9} \cdot \frac{4}{3}$

Step 5.12.3.13

Multiply $\frac{13}{9}$ by $\frac{4}{3}$ .

$\frac{13 \cdot 4}{9 \cdot 3}$

Step 5.12.3.14

Multiply $13$ by $4$ .

$\frac{52}{9 \cdot 3}$

Step 5.12.3.15

Multiply $9$ by $3$ .

$\frac{52}{27}$

Step 6