Calculus Examples

$x = 2 y^{2}$ , $x = 0$ , $y = 2$

Step 1

Solve by substitution to find the intersection between the curves.

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Step 1.1

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

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Step 1.1.1

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

$x = 2 {(2)}^{2}$

$y = 2$

Step 1.1.2

Simplify the right side.

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Step 1.1.2.1

Simplify $2 {(2)}^{2}$ .

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Step 1.1.2.1.1

Multiply $2$ by ${(2)}^{2}$ by adding the exponents.

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Step 1.1.2.1.1.1

Multiply $2$ by ${(2)}^{2}$ .

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Step 1.1.2.1.1.1.1

Raise $2$ to the power of $1$ .

$x = 2 {(2)}^{2}$

$y = 2$

Step 1.1.2.1.1.1.2

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

$x = 2^{1 + 2}$

$y = 2$

$x = 2^{1 + 2}$

$y = 2$

Step 1.1.2.1.1.2

Add $1$ and $2$ .

$x = 2^{3}$

$y = 2$

$x = 2^{3}$

$y = 2$

Step 1.1.2.1.2

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

$x = 8$

$y = 2$

$x = 8$

$y = 2$

$x = 8$

$y = 2$

$x = 8$

$y = 2$

Step 1.2

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

$(8, 2)$

Step 2

The area between the given curves is unbounded.

Unbounded area

Step 3