Algebra Examples

$y = \sqrt[3]{x} + 1$ ; $[0, 1]$

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.

$\sqrt[3]{x} + 1 = 0$

Step 1.2

Solve $\sqrt[3]{x} + 1 = 0$ for $x$ .

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Step 1.2.1

Subtract $1$ from both sides of the equation.

$\sqrt[3]{x} = - 1$

Step 1.2.2

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

${\sqrt[3]{x}}^{3} = {(- 1)}^{3}$

Step 1.2.3

Simplify each side of the equation.

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Step 1.2.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

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

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

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

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Step 1.2.3.2.1.1

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

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Step 1.2.3.2.1.1.1

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

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

Step 1.2.3.2.1.1.2

Cancel the common factor of $3$ .

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Step 1.2.3.2.1.1.2.1

Cancel the common factor.

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

Step 1.2.3.2.1.1.2.2

Rewrite the expression.

$x^{1} = {(- 1)}^{3}$

Step 1.2.3.2.1.2

Simplify.

$x = {(- 1)}^{3}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Raise $- 1$ to the power of $3$ .

$x = - 1$

Step 1.3

Substitute $- 1$ for $x$ .

$y = 0$

Step 1.4

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

$(- 1, 0)$

Step 2

The area of the region between the curves is defined as the integral of the upper curve minus the integral of the lower curve over each region. The regions are determined by the intersection points of the curves. This can be done algebraically or graphically.

$A r e a = \int_{0}^{1} \sqrt[3]{x} + 1 d x - \int_{0}^{1} 0 d x$

Step 3

Integrate to find the area between $0$ and $1$ .

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Step 3.1

Combine the integrals into a single integral.

$\int_{0}^{1} \sqrt[3]{x} + 1 - (0) d x$

Step 3.2

Subtract $0$ from $\sqrt[3]{x} + 1$ .

$\int_{0}^{1} \sqrt[3]{x} + 1 d x$

Step 3.3

Split the single integral into multiple integrals.

$\int_{0}^{1} \sqrt[3]{x} d x + \int_{0}^{1} d x$

Step 3.4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

$\int_{0}^{1} x^{\frac{1}{3}} d x + \int_{0}^{1} d x$

Step 3.5

By the Power Rule, the integral of $x^{\frac{1}{3}}$ with respect to $x$ is $\frac{3}{4} x^{\frac{4}{3}}$ .

$\frac{3}{4} x^{\frac{4}{3}}]_{0}^{1} + \int_{0}^{1} d x$

Step 3.6

Apply the constant rule.

$\frac{3}{4} x^{\frac{4}{3}}]_{0}^{1} + x]_{0}^{1}$

Step 3.7

Simplify the answer.

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Step 3.7.1

Combine $\frac{3}{4} x^{\frac{4}{3}}]_{0}^{1}$ and $x]_{0}^{1}$ .

$\frac{3}{4} x^{\frac{4}{3}} + x]_{0}^{1}$

Step 3.7.2

Substitute and simplify.

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Step 3.7.2.1

Evaluate $\frac{3}{4} x^{\frac{4}{3}} + x$ at $1$ and at $0$ .

$(\frac{3}{4} \cdot 1^{\frac{4}{3}} + 1) - (\frac{3}{4} \cdot 0^{\frac{4}{3}} + 0)$

Step 3.7.2.2

Simplify.

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Step 3.7.2.2.1

One to any power is one.

$\frac{3}{4} \cdot 1 + 1 - (\frac{3}{4} \cdot 0^{\frac{4}{3}} + 0)$

Step 3.7.2.2.2

Multiply $\frac{3}{4}$ by $1$ .

$\frac{3}{4} + 1 - (\frac{3}{4} \cdot 0^{\frac{4}{3}} + 0)$

Step 3.7.2.2.3

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

$\frac{3}{4} + \frac{4}{4} - (\frac{3}{4} \cdot 0^{\frac{4}{3}} + 0)$

Step 3.7.2.2.4

Combine the numerators over the common denominator.

$\frac{3 + 4}{4} - (\frac{3}{4} \cdot 0^{\frac{4}{3}} + 0)$

Step 3.7.2.2.5

Add $3$ and $4$ .

$\frac{7}{4} - (\frac{3}{4} \cdot 0^{\frac{4}{3}} + 0)$

Step 3.7.2.2.6

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

$\frac{7}{4} - (\frac{3}{4} \cdot {(0^{3})}^{\frac{4}{3}} + 0)$

Step 3.7.2.2.7

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

$\frac{7}{4} - (\frac{3}{4} \cdot 0^{3 (\frac{4}{3})} + 0)$

Step 3.7.2.2.8

Cancel the common factor of $3$ .

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Step 3.7.2.2.8.1

Cancel the common factor.

$\frac{7}{4} - (\frac{3}{4} \cdot 0^{3 (\frac{4}{3})} + 0)$

Step 3.7.2.2.8.2

Rewrite the expression.

$\frac{7}{4} - (\frac{3}{4} \cdot 0^{4} + 0)$

Step 3.7.2.2.9

Raising $0$ to any positive power yields $0$ .

$\frac{7}{4} - (\frac{3}{4} \cdot 0 + 0)$

Step 3.7.2.2.10

Multiply $\frac{3}{4}$ by $0$ .

$\frac{7}{4} - (0 + 0)$

Step 3.7.2.2.11

Add $0$ and $0$ .

$\frac{7}{4} - 0$

Step 3.7.2.2.12

Multiply $- 1$ by $0$ .

$\frac{7}{4} + 0$

Step 3.7.2.2.13

Add $\frac{7}{4}$ and $0$ .

$\frac{7}{4}$

Step 4