Calculus Examples

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

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.

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

Step 1.2

Solve $x^{2} = \frac{x^{2}}{2}$ for $x$ .

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Step 1.2.1

Multiply both sides by $2$ .

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

Step 1.2.2

Simplify.

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Step 1.2.2.1

Simplify the left side.

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Step 1.2.2.1.1

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

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

Step 1.2.2.2

Simplify the right side.

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Step 1.2.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.2.1.1

Cancel the common factor.

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

Step 1.2.2.2.1.2

Rewrite the expression.

$2 x^{2} = x^{2}$

Step 1.2.3

Solve for $x$ .

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Step 1.2.3.1

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

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Step 1.2.3.1.1

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

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

Step 1.2.3.1.2

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

$x^{2} = 0$

Step 1.2.3.2

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

$x = \pm \sqrt{0}$

Step 1.2.3.3

Simplify $\pm \sqrt{0}$ .

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Step 1.2.3.3.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 1.2.3.3.2

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

$x = \pm 0$

Step 1.2.3.3.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 1.3

Evaluate $y$ when $x = 0$ .

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Step 1.3.1

Substitute $0$ for $x$ .

$y = \frac{{(0)}^{2}}{2}$

Step 1.3.2

Substitute $0$ for $x$ in $y = \frac{{(0)}^{2}}{2}$ and solve for $y$ .

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Step 1.3.2.1

Remove parentheses.

$y = \frac{0^{2}}{2}$

Step 1.3.2.2

Simplify $\frac{0^{2}}{2}$ .

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Step 1.3.2.2.1

Raising $0$ to any positive power yields $0$ .

$y = \frac{0}{2}$

Step 1.3.2.2.2

Divide $0$ by $2$ .

$y = 0$

Step 1.4

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

$(0, 0)$

Step 2

The area between the given curves is unbounded.

Unbounded area

Step 3