Calculus Examples

$r = 4 \sin (θ) + 4 \cos (θ)$

Step 1

Since $\sin (θ) = \frac{y}{r}$ , replace $\sin (θ)$ with $\frac{y}{r}$ .

$r = 4 \frac{y}{r} + 4 \cos (θ)$

Step 2

Since $\cos (θ) = \frac{x}{r}$ , replace $\cos (θ)$ with $\frac{x}{r}$ .

$r = 4 \frac{y}{r} + 4 \frac{x}{r}$

Step 3

Multiply both sides by $r$ .

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Step 3.1

Multiply both sides by $r$ .

$r \cdot r = r (4 \frac{y}{r} + 4 \frac{x}{r})$

Step 3.2

Simplify the left side.

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Step 3.2.1

Multiply $r$ by $r$ .

$r^{2} = r (4 \frac{y}{r} + 4 \frac{x}{r})$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify $r (4 \frac{y}{r} + 4 \frac{x}{r})$ .

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Step 3.3.1.1

Simplify each term.

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Step 3.3.1.1.1

Combine $4$ and $\frac{y}{r}$ .

$r^{2} = r (\frac{4 y}{r} + 4 \frac{x}{r})$

Step 3.3.1.1.2

Combine $4$ and $\frac{x}{r}$ .

$r^{2} = r (\frac{4 y}{r} + \frac{4 x}{r})$

Step 3.3.1.2

Simplify terms.

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Step 3.3.1.2.1

Apply the distributive property.

$r^{2} = r \frac{4 y}{r} + r \frac{4 x}{r}$

Step 3.3.1.2.2

Cancel the common factor of $r$ .

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Step 3.3.1.2.2.1

Cancel the common factor.

$r^{2} = r \frac{4 y}{r} + r \frac{4 x}{r}$

Step 3.3.1.2.2.2

Rewrite the expression.

$r^{2} = 4 y + r \frac{4 x}{r}$

Step 3.3.1.2.3

Cancel the common factor of $r$ .

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Step 3.3.1.2.3.1

Cancel the common factor.

$r^{2} = 4 y + r \frac{4 x}{r}$

Step 3.3.1.2.3.2

Rewrite the expression.

$r^{2} = 4 y + 4 x$

Step 4

Since $r^{2} = x^{2} + y^{2}$ , replace $r^{2}$ with $x^{2} + y^{2}$ and $r$ with $\sqrt{x^{2} + y^{2}}$ .

$x^{2} + y^{2} = 4 y + 4 x$

Step 5

Write in standard form.

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Step 5.1

Move all terms containing variables to the left side of the equation.

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Step 5.1.1

Subtract $4 y$ from both sides of the equation.

$x^{2} + y^{2} - 4 y = 4 x$

Step 5.1.2

Subtract $4 x$ from both sides of the equation.

$x^{2} + y^{2} - 4 y - 4 x = 0$

Step 5.1.3

Move $- 4 y$ .

$x^{2} + y^{2} - 4 x - 4 y = 0$

Step 5.2

Complete the square for $x^{2} - 4 x$ .

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Step 5.2.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 1$

$b = - 4$

$c = 0$

Step 5.2.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 5.2.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 5.2.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- 4}{2 \cdot 1}$

Step 5.2.3.2

Cancel the common factor of $- 4$ and $2$ .

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Step 5.2.3.2.1

Factor $2$ out of $- 4$ .

$d = \frac{2 \cdot - 2}{2 \cdot 1}$

Step 5.2.3.2.2

Cancel the common factors.

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Step 5.2.3.2.2.1

Factor $2$ out of $2 \cdot 1$ .

$d = \frac{2 \cdot - 2}{2 (1)}$

Step 5.2.3.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 2}{2 \cdot 1}$

Step 5.2.3.2.2.3

Rewrite the expression.

$d = \frac{- 2}{1}$

Step 5.2.3.2.2.4

Divide $- 2$ by $1$ .

$d = - 2$

Step 5.2.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 5.2.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 4)}^{2}}{4 \cdot 1}$

Step 5.2.4.2

Simplify the right side.

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Step 5.2.4.2.1

Simplify each term.

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Step 5.2.4.2.1.1

Cancel the common factor of ${(- 4)}^{2}$ and $4$ .

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Step 5.2.4.2.1.1.1

Rewrite $- 4$ as $- 1 (4)$ .

$e = 0 - \frac{{(- 1 (4))}^{2}}{4 \cdot 1}$

Step 5.2.4.2.1.1.2

Apply the product rule to $- 1 (4)$ .

$e = 0 - \frac{{(- 1)}^{2} \cdot 4^{2}}{4 \cdot 1}$

Step 5.2.4.2.1.1.3

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

$e = 0 - \frac{1 \cdot 4^{2}}{4 \cdot 1}$

Step 5.2.4.2.1.1.4

Multiply $4^{2}$ by $1$ .

$e = 0 - \frac{4^{2}}{4 \cdot 1}$

Step 5.2.4.2.1.1.5

Factor $4$ out of $4^{2}$ .

$e = 0 - \frac{4 \cdot 4}{4 \cdot 1}$

Step 5.2.4.2.1.1.6

Cancel the common factors.

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Step 5.2.4.2.1.1.6.1

Factor $4$ out of $4 \cdot 1$ .

$e = 0 - \frac{4 \cdot 4}{4 (1)}$

Step 5.2.4.2.1.1.6.2

Cancel the common factor.

$e = 0 - \frac{4 \cdot 4}{4 \cdot 1}$

Step 5.2.4.2.1.1.6.3

Rewrite the expression.

$e = 0 - \frac{4}{1}$

Step 5.2.4.2.1.1.6.4

Divide $4$ by $1$ .

$e = 0 - 1 \cdot 4$

Step 5.2.4.2.1.2

Multiply $- 1$ by $4$ .

$e = 0 - 4$

Step 5.2.4.2.2

Subtract $4$ from $0$ .

$e = - 4$

Step 5.2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(x - 2)}^{2} - 4$ .

${(x - 2)}^{2} - 4$

Step 5.3

Substitute ${(x - 2)}^{2} - 4$ for $x^{2} - 4 x$ in the equation $x^{2} + y^{2} - 4 x - 4 y = 0$ .

${(x - 2)}^{2} - 4 + y^{2} - 4 y = 0$

Step 5.4

Move $- 4$ to the right side of the equation by adding $4$ to both sides.

${(x - 2)}^{2} + y^{2} - 4 y = 0 + 4$

Step 5.5

Complete the square for $y^{2} - 4 y$ .

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Step 5.5.1

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = 1$

$b = - 4$

$c = 0$

Step 5.5.2

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 5.5.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 5.5.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- 4}{2 \cdot 1}$

Step 5.5.3.2

Cancel the common factor of $- 4$ and $2$ .

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Step 5.5.3.2.1

Factor $2$ out of $- 4$ .

$d = \frac{2 \cdot - 2}{2 \cdot 1}$

Step 5.5.3.2.2

Cancel the common factors.

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Step 5.5.3.2.2.1

Factor $2$ out of $2 \cdot 1$ .

$d = \frac{2 \cdot - 2}{2 (1)}$

Step 5.5.3.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot - 2}{2 \cdot 1}$

Step 5.5.3.2.2.3

Rewrite the expression.

$d = \frac{- 2}{1}$

Step 5.5.3.2.2.4

Divide $- 2$ by $1$ .

$d = - 2$

Step 5.5.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 5.5.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 0 - \frac{{(- 4)}^{2}}{4 \cdot 1}$

Step 5.5.4.2

Simplify the right side.

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Step 5.5.4.2.1

Simplify each term.

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Step 5.5.4.2.1.1

Cancel the common factor of ${(- 4)}^{2}$ and $4$ .

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Step 5.5.4.2.1.1.1

Rewrite $- 4$ as $- 1 (4)$ .

$e = 0 - \frac{{(- 1 (4))}^{2}}{4 \cdot 1}$

Step 5.5.4.2.1.1.2

Apply the product rule to $- 1 (4)$ .

$e = 0 - \frac{{(- 1)}^{2} \cdot 4^{2}}{4 \cdot 1}$

Step 5.5.4.2.1.1.3

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

$e = 0 - \frac{1 \cdot 4^{2}}{4 \cdot 1}$

Step 5.5.4.2.1.1.4

Multiply $4^{2}$ by $1$ .

$e = 0 - \frac{4^{2}}{4 \cdot 1}$

Step 5.5.4.2.1.1.5

Factor $4$ out of $4^{2}$ .

$e = 0 - \frac{4 \cdot 4}{4 \cdot 1}$

Step 5.5.4.2.1.1.6

Cancel the common factors.

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Step 5.5.4.2.1.1.6.1

Factor $4$ out of $4 \cdot 1$ .

$e = 0 - \frac{4 \cdot 4}{4 (1)}$

Step 5.5.4.2.1.1.6.2

Cancel the common factor.

$e = 0 - \frac{4 \cdot 4}{4 \cdot 1}$

Step 5.5.4.2.1.1.6.3

Rewrite the expression.

$e = 0 - \frac{4}{1}$

Step 5.5.4.2.1.1.6.4

Divide $4$ by $1$ .

$e = 0 - 1 \cdot 4$

Step 5.5.4.2.1.2

Multiply $- 1$ by $4$ .

$e = 0 - 4$

Step 5.5.4.2.2

Subtract $4$ from $0$ .

$e = - 4$

Step 5.5.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form ${(y - 2)}^{2} - 4$ .

${(y - 2)}^{2} - 4$

Step 5.6

Substitute ${(y - 2)}^{2} - 4$ for $y^{2} - 4 y$ in the equation $x^{2} + y^{2} - 4 x - 4 y = 0$ .

${(x - 2)}^{2} + {(y - 2)}^{2} - 4 = 0 + 4$

Step 5.7

Move $- 4$ to the right side of the equation by adding $4$ to both sides.

${(x - 2)}^{2} + {(y - 2)}^{2} = 0 + 4 + 4$

Step 5.8

Simplify $0 + 4 + 4$ .

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Step 5.8.1

Add $0$ and $4$ .

${(x - 2)}^{2} + {(y - 2)}^{2} = 4 + 4$

Step 5.8.2

Add $4$ and $4$ .

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