Calculus Examples

$y = \frac{1}{2} x^{2}$ , $y = - x^{2} + 6$

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.

$\frac{1}{2} x^{2} = - x^{2} + 6$

Step 1.2

Solve $\frac{1}{2} x^{2} = - x^{2} + 6$ for $x$ .

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Step 1.2.1

Simplify $\frac{1}{2} x^{2}$ .

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Step 1.2.1.1

Rewrite.

$0 + 0 + \frac{1}{2} x^{2} = - x^{2} + 6$

Step 1.2.1.2

Simplify by adding zeros.

$\frac{1}{2} x^{2} = - x^{2} + 6$

Step 1.2.1.3

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

$\frac{x^{2}}{2} = - x^{2} + 6$

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.

$\frac{x^{2}}{2} + x^{2} = 6$

Step 1.2.2.2

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

$\frac{x^{2}}{2} + x^{2} \cdot \frac{2}{2} = 6$

Step 1.2.2.3

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

$\frac{x^{2}}{2} + \frac{x^{2} \cdot 2}{2} = 6$

Step 1.2.2.4

Combine the numerators over the common denominator.

$\frac{x^{2} + x^{2} \cdot 2}{2} = 6$

Step 1.2.2.5

Simplify the numerator.

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Step 1.2.2.5.1

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

$\frac{x^{2} + 2 \cdot x^{2}}{2} = 6$

Step 1.2.2.5.2

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

$\frac{3 x^{2}}{2} = 6$

Step 1.2.3

Multiply both sides of the equation by $\frac{2}{3}$ .

$\frac{2}{3} \cdot \frac{3 x^{2}}{2} = \frac{2}{3} \cdot 6$

Step 1.2.4

Simplify both sides of the equation.

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Step 1.2.4.1

Simplify the left side.

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Step 1.2.4.1.1

Simplify $\frac{2}{3} \cdot \frac{3 x^{2}}{2}$ .

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Step 1.2.4.1.1.1

Combine.

$\frac{2 (3 x^{2})}{3 \cdot 2} = \frac{2}{3} \cdot 6$

Step 1.2.4.1.1.2

Cancel the common factor of $2$ .

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Step 1.2.4.1.1.2.1

Cancel the common factor.

$\frac{2 (3 x^{2})}{3 \cdot 2} = \frac{2}{3} \cdot 6$

Step 1.2.4.1.1.2.2

Rewrite the expression.

$\frac{3 x^{2}}{3} = \frac{2}{3} \cdot 6$

Step 1.2.4.1.1.3

Cancel the common factor of $3$ .

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Step 1.2.4.1.1.3.1

Cancel the common factor.

$\frac{3 x^{2}}{3} = \frac{2}{3} \cdot 6$

Step 1.2.4.1.1.3.2

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

$x^{2} = \frac{2}{3} \cdot 6$

Step 1.2.4.2

Simplify the right side.

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Step 1.2.4.2.1

Simplify $\frac{2}{3} \cdot 6$ .

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Step 1.2.4.2.1.1

Cancel the common factor of $3$ .

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Step 1.2.4.2.1.1.1

Factor $3$ out of $6$ .

$x^{2} = \frac{2}{3} \cdot (3 (2))$

Step 1.2.4.2.1.1.2

Cancel the common factor.

$x^{2} = \frac{2}{3} \cdot (3 \cdot 2)$

Step 1.2.4.2.1.1.3

Rewrite the expression.

$x^{2} = 2 \cdot 2$

Step 1.2.4.2.1.2

Multiply $2$ by $2$ .

$x^{2} = 4$

Step 1.2.5

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

$x = \pm \sqrt{4}$

Step 1.2.6

Simplify $\pm \sqrt{4}$ .

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Step 1.2.6.1

Rewrite $4$ as $2^{2}$ .

$x = \pm \sqrt{2^{2}}$

Step 1.2.6.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 2$

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.

$x = 2$

Step 1.2.7.2

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

$x = - 2$

Step 1.2.7.3

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

$x = 2, - 2$

Step 1.3

Evaluate $y$ when $x = 2$ .

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Step 1.3.1

Substitute $2$ for $x$ .

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

Step 1.3.2

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

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Step 1.3.2.1

Remove parentheses.

$y = - 2^{2} + 6$

Step 1.3.2.2

Simplify $- 2^{2} + 6$ .

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Step 1.3.2.2.1

Simplify each term.

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Step 1.3.2.2.1.1

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

$y = - 1 \cdot 4 + 6$

Step 1.3.2.2.1.2

Multiply $- 1$ by $4$ .

$y = - 4 + 6$

Step 1.3.2.2.2

Add $- 4$ and $6$ .

$y = 2$

Step 1.4

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

$(2, 2)$

$(- 2, 2)$

$(2, 2)$

$(- 2, 2)$

Step 2

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

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

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.

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

Step 4

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

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Step 4.1

Combine the integrals into a single integral.

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

Step 4.2

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

$\int_{- 2}^{2} - x^{2} \cdot \frac{2}{2} - \frac{x^{2}}{2} + 6 d x$

Step 4.3

Simplify terms.

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Step 4.3.1

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

$\frac{- x^{2} \cdot 2}{2} - \frac{x^{2}}{2} + 6$

Step 4.3.2

Combine the numerators over the common denominator.

$\frac{- x^{2} \cdot 2 - x^{2}}{2} + 6$

$\int_{- 2}^{2} \frac{- x^{2} \cdot 2 - x^{2}}{2} + 6 d x$

Step 4.4

Simplify each term.

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Step 4.4.1

Simplify the numerator.

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Step 4.4.1.1

Factor $x^{2}$ out of $- x^{2} \cdot 2 - x^{2}$ .

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Step 4.4.1.1.1

Factor $x^{2}$ out of $- x^{2} \cdot 2$ .

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

Step 4.4.1.1.2

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

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

Step 4.4.1.1.3

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

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

Step 4.4.1.2

Multiply $- 1$ by $2$ .

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

Step 4.4.1.3

Subtract $1$ from $- 2$ .

$\frac{x^{2} \cdot - 3}{2} + 6$

Step 4.4.2

Move $- 3$ to the left of $x^{2}$ .

$\frac{- 3 \cdot x^{2}}{2} + 6$

Step 4.4.3

Move the negative in front of the fraction.

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

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

Step 4.5

Split the single integral into multiple integrals.

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

Step 4.6

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

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

Step 4.7

Since $\frac{3}{2}$ is constant with respect to $x$ , move $\frac{3}{2}$ out of the integral.

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

Step 4.8

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

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

Step 4.9

Apply the constant rule.

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

Step 4.10

Substitute and simplify.

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Step 4.10.1

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

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

Step 4.10.2

Evaluate $6 x$ at $2$ and at $- 2$ .

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

Step 4.10.3

Simplify.

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Step 4.10.3.1

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

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

Step 4.10.3.2

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

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

Step 4.10.3.3

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

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

Step 4.10.3.4

Multiply $- 8$ by $- 1$ .

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

Step 4.10.3.5

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

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

Step 4.10.3.6

Combine the numerators over the common denominator.

$- \frac{3}{2} \cdot \frac{8 + 8}{3} + 6 \cdot 2 - 6 \cdot - 2$

Step 4.10.3.7

Add $8$ and $8$ .

$- \frac{3}{2} \cdot \frac{16}{3} + 6 \cdot 2 - 6 \cdot - 2$

Step 4.10.3.8

Multiply $\frac{16}{3}$ by $\frac{3}{2}$ .

$- \frac{16 \cdot 3}{3 \cdot 2} + 6 \cdot 2 - 6 \cdot - 2$

Step 4.10.3.9

Multiply $16$ by $3$ .

$- \frac{48}{3 \cdot 2} + 6 \cdot 2 - 6 \cdot - 2$

Step 4.10.3.10

Multiply $3$ by $2$ .

$- \frac{48}{6} + 6 \cdot 2 - 6 \cdot - 2$

Step 4.10.3.11

Cancel the common factor of $48$ and $6$ .

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Step 4.10.3.11.1

Factor $6$ out of $48$ .

$- \frac{6 \cdot 8}{6} + 6 \cdot 2 - 6 \cdot - 2$

Step 4.10.3.11.2

Cancel the common factors.

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Step 4.10.3.11.2.1

Factor $6$ out of $6$ .

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

Step 4.10.3.11.2.2

Cancel the common factor.

$- \frac{6 \cdot 8}{6 \cdot 1} + 6 \cdot 2 - 6 \cdot - 2$

Step 4.10.3.11.2.3

Rewrite the expression.

$- \frac{8}{1} + 6 \cdot 2 - 6 \cdot - 2$

Step 4.10.3.11.2.4

Divide $8$ by $1$ .

$- 1 \cdot 8 + 6 \cdot 2 - 6 \cdot - 2$

Step 4.10.3.12

Multiply $- 1$ by $8$ .

$- 8 + 6 \cdot 2 - 6 \cdot - 2$

Step 4.10.3.13

Multiply $6$ by $2$ .

$- 8 + 12 - 6 \cdot - 2$

Step 4.10.3.14

Multiply $- 6$ by $- 2$ .

$- 8 + 12 + 12$

Step 4.10.3.15

Add $12$ and $12$ .

$- 8 + 24$

Step 4.10.3.16

Add $- 8$ and $24$ .

$16$

Step 5