Calculus Examples

$y = x^{\frac{4}{3}}$ , $y = 2 x^{\frac{1}{3}}$

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.

$x^{\frac{4}{3}} = 2 x^{\frac{1}{3}}$

Step 1.2

Solve $x^{\frac{4}{3}} = 2 x^{\frac{1}{3}}$ for $x$ .

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Step 1.2.1

Eliminate the fractional exponents by multiplying both exponents by the LCD.

${(x^{\frac{4}{3}})}^{3} = {(2 x^{\frac{1}{3}})}^{3}$

Step 1.2.2

Multiply the exponents in ${(x^{\frac{4}{3}})}^{3}$ .

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Step 1.2.2.1

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

$x^{\frac{4}{3} \cdot 3} = {(2 x^{\frac{1}{3}})}^{3}$

Step 1.2.2.2

Cancel the common factor of $3$ .

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Step 1.2.2.2.1

Cancel the common factor.

$x^{\frac{4}{3} \cdot 3} = {(2 x^{\frac{1}{3}})}^{3}$

Step 1.2.2.2.2

Rewrite the expression.

$x^{4} = {(2 x^{\frac{1}{3}})}^{3}$

Step 1.2.3

Simplify ${(2 x^{\frac{1}{3}})}^{3}$ .

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Step 1.2.3.1

Apply the product rule to $2 x^{\frac{1}{3}}$ .

$x^{4} = 2^{3} {(x^{\frac{1}{3}})}^{3}$

Step 1.2.3.2

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

$x^{4} = 8 {(x^{\frac{1}{3}})}^{3}$

Step 1.2.3.3

Multiply the exponents in ${(x^{\frac{1}{3}})}^{3}$ .

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Step 1.2.3.3.1

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

$x^{4} = 8 x^{\frac{1}{3} \cdot 3}$

Step 1.2.3.3.2

Cancel the common factor of $3$ .

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Step 1.2.3.3.2.1

Cancel the common factor.

$x^{4} = 8 x^{\frac{1}{3} \cdot 3}$

Step 1.2.3.3.2.2

Rewrite the expression.

$x^{4} = 8 x^{1}$

Step 1.2.3.4

Simplify.

$x^{4} = 8 x$

Step 1.2.4

Subtract $8 x$ from both sides of the equation.

$x^{4} - 8 x = 0$

Step 1.2.5

Factor the left side of the equation.

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Step 1.2.5.1

Factor $x$ out of $x^{4} - 8 x$ .

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Step 1.2.5.1.1

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

$x \cdot x^{3} - 8 x = 0$

Step 1.2.5.1.2

Factor $x$ out of $- 8 x$ .

$x \cdot x^{3} + x \cdot - 8 = 0$

Step 1.2.5.1.3

Factor $x$ out of $x \cdot x^{3} + x \cdot - 8$ .

$x (x^{3} - 8) = 0$

Step 1.2.5.2

Rewrite $8$ as $2^{3}$ .

$x (x^{3} - 2^{3}) = 0$

Step 1.2.5.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 2$ .

$x ((x - 2) (x^{2} + x \cdot 2 + 2^{2})) = 0$

Step 1.2.5.4

Factor.

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Step 1.2.5.4.1

Simplify.

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Step 1.2.5.4.1.1

Move $2$ to the left of $x$ .

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

Step 1.2.5.4.1.2

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

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

Step 1.2.5.4.2

Remove unnecessary parentheses.

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

Step 1.2.6

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 2 = 0$

$x^{2} + 2 x + 4 = 0 + y = 2 x^{\frac{1}{3}}$

Step 1.2.7

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.8

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 1.2.8.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 1.2.8.2

Add $2$ to both sides of the equation.

$x = 2$

Step 1.2.9

Set $x^{2} + 2 x + 4$ equal to $0$ and solve for $x$ .

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Step 1.2.9.1

Set $x^{2} + 2 x + 4$ equal to $0$ .

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

Step 1.2.9.2

Solve $x^{2} + 2 x + 4 = 0$ for $x$ .

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Step 1.2.9.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a} + y = 2 x^{\frac{1}{3}}$

Step 1.2.9.2.2

Substitute the values $a = 1$ , $b = 2$ , and $c = 4$ into the quadratic formula and solve for $x$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (1 \cdot 4)}}{2 \cdot 1} + y = 2 x^{\frac{1}{3}}$

Step 1.2.9.2.3

Simplify.

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Step 1.2.9.2.3.1

Simplify the numerator.

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Step 1.2.9.2.3.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Step 1.2.9.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 4$ .

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Step 1.2.9.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 4}}{2 \cdot 1}$

Step 1.2.9.2.3.1.2.2

Multiply $- 4$ by $4$ .

$x = \frac{- 2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

Step 1.2.9.2.3.1.3

Subtract $16$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 12}}{2 \cdot 1}$

Step 1.2.9.2.3.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 1}$

Step 1.2.9.2.3.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 1}$

Step 1.2.9.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

Step 1.2.9.2.3.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 1.2.9.2.3.1.7.1

Factor $4$ out of $12$ .

$x = \frac{- 2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 1}$

Step 1.2.9.2.3.1.7.2

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

$x = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 1.2.9.2.3.1.8

Pull terms out from under the radical.

$x = \frac{- 2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 1}$

Step 1.2.9.2.3.1.9

Move $2$ to the left of $i$ .

$x = \frac{- 2 \pm 2 i \sqrt{3}}{2 \cdot 1}$

Step 1.2.9.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 i \sqrt{3}}{2}$

Step 1.2.9.2.3.3

Simplify $\frac{- 2 \pm 2 i \sqrt{3}}{2}$ .

$x = - 1 \pm i \sqrt{3}$

Step 1.2.9.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 1.2.9.2.4.1

Simplify the numerator.

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Step 1.2.9.2.4.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Step 1.2.9.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 4$ .

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Step 1.2.9.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 4}}{2 \cdot 1}$

Step 1.2.9.2.4.1.2.2

Multiply $- 4$ by $4$ .

$x = \frac{- 2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

Step 1.2.9.2.4.1.3

Subtract $16$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 12}}{2 \cdot 1}$

Step 1.2.9.2.4.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 1}$

Step 1.2.9.2.4.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 1}$

Step 1.2.9.2.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

Step 1.2.9.2.4.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 1.2.9.2.4.1.7.1

Factor $4$ out of $12$ .

$x = \frac{- 2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 1}$

Step 1.2.9.2.4.1.7.2

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

$x = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 1.2.9.2.4.1.8

Pull terms out from under the radical.

$x = \frac{- 2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 1}$

Step 1.2.9.2.4.1.9

Move $2$ to the left of $i$ .

$x = \frac{- 2 \pm 2 i \sqrt{3}}{2 \cdot 1}$

Step 1.2.9.2.4.2

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 i \sqrt{3}}{2}$

Step 1.2.9.2.4.3

Simplify $\frac{- 2 \pm 2 i \sqrt{3}}{2}$ .

$x = - 1 \pm i \sqrt{3}$

Step 1.2.9.2.4.4

Change the $\pm$ to $+$ .

$x = - 1 + i \sqrt{3}$

Step 1.2.9.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 1.2.9.2.5.1

Simplify the numerator.

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Step 1.2.9.2.5.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot 4}}{2 \cdot 1}$

Step 1.2.9.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 4$ .

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Step 1.2.9.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 4}}{2 \cdot 1}$

Step 1.2.9.2.5.1.2.2

Multiply $- 4$ by $4$ .

$x = \frac{- 2 \pm \sqrt{4 - 16}}{2 \cdot 1}$

Step 1.2.9.2.5.1.3

Subtract $16$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 12}}{2 \cdot 1}$

Step 1.2.9.2.5.1.4

Rewrite $- 12$ as $- 1 (12)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 12}}{2 \cdot 1}$

Step 1.2.9.2.5.1.5

Rewrite $\sqrt{- 1 (12)}$ as $\sqrt{- 1} \cdot \sqrt{12}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{12}}{2 \cdot 1}$

Step 1.2.9.2.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 2 \pm i \cdot \sqrt{12}}{2 \cdot 1}$

Step 1.2.9.2.5.1.7

Rewrite $12$ as $2^{2} \cdot 3$ .

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Step 1.2.9.2.5.1.7.1

Factor $4$ out of $12$ .

$x = \frac{- 2 \pm i \cdot \sqrt{4 (3)}}{2 \cdot 1}$

Step 1.2.9.2.5.1.7.2

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

$x = \frac{- 2 \pm i \cdot \sqrt{2^{2} \cdot 3}}{2 \cdot 1}$

Step 1.2.9.2.5.1.8

Pull terms out from under the radical.

$x = \frac{- 2 \pm i \cdot (2 \sqrt{3})}{2 \cdot 1}$

Step 1.2.9.2.5.1.9

Move $2$ to the left of $i$ .

$x = \frac{- 2 \pm 2 i \sqrt{3}}{2 \cdot 1}$

Step 1.2.9.2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 i \sqrt{3}}{2}$

Step 1.2.9.2.5.3

Simplify $\frac{- 2 \pm 2 i \sqrt{3}}{2}$ .

$x = - 1 \pm i \sqrt{3}$

Step 1.2.9.2.5.4

Change the $\pm$ to $-$ .

$x = - 1 - i \sqrt{3}$

Step 1.2.9.2.6

The final answer is the combination of both solutions.

$x = - 1 + i \sqrt{3}, - 1 - i \sqrt{3}$

Step 1.2.10

The final solution is all the values that make $x (x - 2) (x^{2} + 2 x + 4) = 0$ true.

$x = 0, 2, - 1 + i \sqrt{3}, - 1 - i \sqrt{3}$

Step 1.3

Evaluate $y$ when $x = 0$ .

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Step 1.3.1

Substitute $0$ for $x$ .

$y = 2 {(0)}^{\frac{1}{3}}$

Step 1.3.2

Substitute $0$ for $x$ in $y = 2 {(0)}^{\frac{1}{3}}$ and solve for $y$ .

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Step 1.3.2.1

Remove parentheses.

$y = 2 \cdot 0^{\frac{1}{3}}$

Step 1.3.2.2

Simplify $2 \cdot 0^{\frac{1}{3}}$ .

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Step 1.3.2.2.1

Simplify the expression.

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Step 1.3.2.2.1.1

Rewrite $0$ as $0^{3}$ .

$y = 2 \cdot {(0^{3})}^{\frac{1}{3}}$

Step 1.3.2.2.1.2

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

$y = 2 \cdot 0^{3 (\frac{1}{3})}$

Step 1.3.2.2.2

Cancel the common factor of $3$ .

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Step 1.3.2.2.2.1

Cancel the common factor.

$y = 2 \cdot 0^{3 (\frac{1}{3})}$

Step 1.3.2.2.2.2

Rewrite the expression.

$y = 2 \cdot 0^{1}$

Step 1.3.2.2.3

Evaluate the exponent.

$y = 2 \cdot 0$

Step 1.3.2.2.4

Multiply $2$ by $0$ .

$y = 0$

Step 1.4

Evaluate $y$ when $x = 2$ .

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Step 1.4.1

Substitute $2$ for $x$ .

$y = 2 {(2)}^{\frac{1}{3}}$

Step 1.4.2

Substitute $2$ for $x$ in $y = 2 {(2)}^{\frac{1}{3}}$ and solve for $y$ .

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Step 1.4.2.1

Remove parentheses.

$y = 2 \cdot 2^{\frac{1}{3}}$

Step 1.4.2.2

Multiply $2$ by $2^{\frac{1}{3}}$ by adding the exponents.

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Step 1.4.2.2.1

Multiply $2$ by $2^{\frac{1}{3}}$ .

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Step 1.4.2.2.1.1

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

$y = 2^{1} \cdot 2^{\frac{1}{3}}$

Step 1.4.2.2.1.2

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

$y = 2^{1 + \frac{1}{3}}$

Step 1.4.2.2.2

Write $1$ as a fraction with a common denominator.

$y = 2^{\frac{3}{3} + \frac{1}{3}}$

Step 1.4.2.2.3

Combine the numerators over the common denominator.

$y = 2^{\frac{3 + 1}{3}}$

Step 1.4.2.2.4

Add $3$ and $1$ .

$y = 2^{\frac{4}{3}}$

Step 1.5

Substitute $- 1 + i \sqrt{3}$ for $x$ .

$y = 2 {(- 1 + i \sqrt{3})}^{\frac{1}{3}}$

Step 1.6

Substitute $- 1 - i \sqrt{3}$ for $x$ .

$y = 2 {(- 1 - i \sqrt{3})}^{\frac{1}{3}}$

Step 1.7

List all of the solutions.

$y = 0, x = 0$

$y = 2^{\frac{4}{3}}, x = 2$

$y = 2 {(- 1 + i \sqrt{3})}^{\frac{1}{3}}, x = - 1 + i \sqrt{3}$

$y = 2 {(- 1 - i \sqrt{3})}^{\frac{1}{3}}, x = - 1 - i \sqrt{3}$

$y = 0, x = 0$

$y = 2^{\frac{4}{3}}, x = 2$

$y = 2 {(- 1 + i \sqrt{3})}^{\frac{1}{3}}, x = - 1 + i \sqrt{3}$

$y = 2 {(- 1 - i \sqrt{3})}^{\frac{1}{3}}, x = - 1 - i \sqrt{3}$

Step 2

The area between the given curves is unbounded.

Unbounded area

Step 3