Calculus Examples

$x = 2 - y^{2}$ , $1 \leq y \leq 2$

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 $1$ in each equation.

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Step 1.1.1

Replace all occurrences of $y$ in $x = 2 - y^{2}$ with $1$ .

$x = 2 - {(1)}^{2}$

$y = 1$

$y = 2$

Step 1.1.2

Simplify the right side.

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Step 1.1.2.1

Simplify $2 - {(1)}^{2}$ .

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Step 1.1.2.1.1

Simplify each term.

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Step 1.1.2.1.1.1

One to any power is one.

$x = 2 - 1 \cdot 1$

$y = 1$

$y = 2$

Step 1.1.2.1.1.2

Multiply $- 1$ by $1$ .

$x = 2 - 1$

$y = 1$

$y = 2$

$x = 2 - 1$

$y = 1$

$y = 2$

Step 1.1.2.1.2

Subtract $1$ from $2$ .

$x = 1$

$y = 1$

$y = 2$

$x = 1$

$y = 1$

$y = 2$

$x = 1$

$y = 1$

$y = 2$

$x = 1$

$y = 1$

$y = 2$

Step 1.2

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

$(1, 1)$

Step 2

The area between the given curves is unbounded.

Unbounded area

Step 3