Calculus Examples

$y = \sqrt{x}$ , $y = \sqrt{8 - x}$ , $y = 0$

Step 1

Eliminate the equal sides of each equation and combine.

$\sqrt{x} = \sqrt{8 - x}$

Step 2

Solve $\sqrt{x} = \sqrt{8 - x}$ for $x$ .

Tap for more steps...

Step 2.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{x}}^{2} = {\sqrt{8 - x}}^{2} + y = \sqrt{8 - x}$

$y = 0$

Step 2.2

Simplify each side of the equation.

Tap for more steps...

Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

${(x^{\frac{1}{2}})}^{2} = {\sqrt{8 - x}}^{2} + y = \sqrt{8 - x}$

$y = 0$

Step 2.2.2

Simplify the left side.

Tap for more steps...

Step 2.2.2.1

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

Tap for more steps...

Step 2.2.2.1.1

Multiply the exponents in ${(x^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 2.2.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{1}{2} \cdot 2} = {\sqrt{8 - x}}^{2} + y = \sqrt{8 - x}$

$y = 0$

Step 2.2.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = {\sqrt{8 - x}}^{2} + y = \sqrt{8 - x}$

$y = 0$

Step 2.2.2.1.1.2.2

Rewrite the expression.

$x = {\sqrt{8 - x}}^{2} + y = \sqrt{8 - x}$

$y = 0$

$x = {\sqrt{8 - x}}^{2} + y = \sqrt{8 - x}$

$y = 0$

$x = {\sqrt{8 - x}}^{2} + y = \sqrt{8 - x}$

$y = 0$

Step 2.2.2.1.2

Simplify.

$x = {\sqrt{8 - x}}^{2} + y = \sqrt{8 - x}$

$y = 0$

$x = {\sqrt{8 - x}}^{2} + y = \sqrt{8 - x}$

$y = 0$

$x = {\sqrt{8 - x}}^{2} + y = \sqrt{8 - x}$

$y = 0$

Step 2.2.3

Simplify the right side.

Tap for more steps...

Step 2.2.3.1

Rewrite ${\sqrt{8 - x}}^{2}$ as $8 - x$ .

Tap for more steps...

Step 2.2.3.1.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{8 - x}$ as ${(8 - x)}^{\frac{1}{2}}$ .

$x = {({(8 - x)}^{\frac{1}{2}})}^{2} + y = \sqrt{8 - x}$

$y = 0$

Step 2.2.3.1.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x = {(8 - x)}^{\frac{1}{2} \cdot 2} + y = \sqrt{8 - x}$

$y = 0$

Step 2.2.3.1.3

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

$x = {(8 - x)}^{\frac{2}{2}} + y = \sqrt{8 - x}$

$y = 0$

Step 2.2.3.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.2.3.1.4.1

Cancel the common factor.

$x = {(8 - x)}^{\frac{2}{2}} + y = \sqrt{8 - x}$

$y = 0$

Step 2.2.3.1.4.2

Rewrite the expression.

$x = (8 - x) + y = \sqrt{8 - x}$

$y = 0$

$x = (8 - x) + y = \sqrt{8 - x}$

$y = 0$

Step 2.2.3.1.5

Simplify.

$x = 8 - x + y = \sqrt{8 - x}$

$y = 0$

$x = 8 - x + y = \sqrt{8 - x}$

$y = 0$

$x = 8 - x + y = \sqrt{8 - x}$

$y = 0$

$x = 8 - x + y = \sqrt{8 - x}$

$y = 0$

Step 2.3

Solve for $x$ .

Tap for more steps...

Step 2.3.1

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

Tap for more steps...

Step 2.3.1.1

Add $x$ to both sides of the equation.

$x + x = 8 + y = \sqrt{8 - x}$

$y = 0$

Step 2.3.1.2

Add $x$ and $x$ .

$2 x = 8 + y = \sqrt{8 - x}$

$y = 0$

$2 x = 8 + y = \sqrt{8 - x}$

$y = 0$

Step 2.3.2

Divide each term in $2 x = 8$ by $2$ and simplify.

Tap for more steps...

Step 2.3.2.1

Divide each term in $2 x = 8$ by $2$ .

$\frac{2 x}{2} = \frac{8}{2} + y = \sqrt{8 - x}$

$y = 0$

Step 2.3.2.2

Simplify the left side.

Tap for more steps...

Step 2.3.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.3.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{8}{2} + y = \sqrt{8 - x}$

$y = 0$

Step 2.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{8}{2} + y = \sqrt{8 - x}$

$y = 0$

$x = \frac{8}{2} + y = \sqrt{8 - x}$

$y = 0$

$x = \frac{8}{2} + y = \sqrt{8 - x}$

$y = 0$

Step 2.3.2.3

Simplify the right side.

Tap for more steps...

Step 2.3.2.3.1

Divide $8$ by $2$ .

$x = 4 + y = \sqrt{8 - x}$

$y = 0$

$x = 4 + y = \sqrt{8 - x}$

$y = 0$

$x = 4 + y = \sqrt{8 - x}$

$y = 0$

$x = 4 + y = \sqrt{8 - x}$

$y = 0$

$x = 4 + y = \sqrt{8 - x}$

$y = 0$

Step 3

Evaluate $y$ when $x = 4$ .

Tap for more steps...

Step 3.1

Substitute $4$ for $x$ .

$y = \sqrt{8 - (4)} y = 0$

Step 3.2

Simplify $\sqrt{8 - (4)}$ .

Tap for more steps...

Step 3.2.1

Multiply $- 1$ by $4$ .

$y = \sqrt{8 - 4}$

Step 3.2.2

Subtract $4$ from $8$ .

$y = \sqrt{4}$

Step 3.2.3

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

$y = \sqrt{2^{2}}$

Step 3.2.4

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

$y = 2$

$y = 2$

$y = 2$

Step 4

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

$(4, 2)$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$