Calculus Examples

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

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.

$8 = \sqrt{x}$

Step 1.2

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

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Step 1.2.1

Rewrite the equation as $\sqrt{x} = 8$ .

$\sqrt{x} = 8$

Step 1.2.2

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

${\sqrt{x}}^{2} = 8^{2}$

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 $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

${(x^{\frac{1}{2}})}^{2} = 8^{2}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

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

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Step 1.2.3.2.1.1

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

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Step 1.2.3.2.1.1.1

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

$x^{\frac{1}{2} \cdot 2} = 8^{2}$

Step 1.2.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 1.2.3.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{2} \cdot 2} = 8^{2}$

Step 1.2.3.2.1.1.2.2

Rewrite the expression.

$x^{1} = 8^{2}$

Step 1.2.3.2.1.2

Simplify.

$x = 8^{2}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Raise $8$ to the power of $2$ .

$x = 64$

Step 1.3

Evaluate $y$ when $x = 64$ .

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Step 1.3.1

Substitute $64$ for $x$ .

$y = \sqrt{64}$

Step 1.3.2

Simplify $\sqrt{64}$ .

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Step 1.3.2.1

Remove parentheses.

$y = \sqrt{64}$

Step 1.3.2.2

Rewrite $64$ as $8^{2}$ .

$y = \sqrt{8^{2}}$

Step 1.3.2.3

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

$y = 8$

Step 1.4

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

$(64, 8)$

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.

$A r e a = \int_{0}^{64} 8 d x - \int_{0}^{64} \sqrt{x} d x$

Step 3

Integrate to find the area between $0$ and $64$ .

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Step 3.1

Integrate to find the area between $0$ and $64$ .

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Step 3.1.1

Combine the integrals into a single integral.

$\int_{0}^{64} \sqrt{x} - (8) d x$

Step 3.1.2

Multiply $- 1$ by $8$ .

$\int_{0}^{64} \sqrt{x} - 8 d x$

Step 3.1.3

Split the single integral into multiple integrals.

$\int_{0}^{64} \sqrt{x} d x + \int_{0}^{64} - 8 d x$

Step 3.1.4

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

$\int_{0}^{64} x^{\frac{1}{2}} d x + \int_{0}^{64} - 8 d x$

Step 3.1.5

By the Power Rule, the integral of $x^{\frac{1}{2}}$ with respect to $x$ is $\frac{2}{3} x^{\frac{3}{2}}$ .

$\frac{2}{3} x^{\frac{3}{2}}]_{0}^{64} + \int_{0}^{64} - 8 d x$

Step 3.1.6

Apply the constant rule.

$\frac{2}{3} x^{\frac{3}{2}}]_{0}^{64} + - 8 x]_{0}^{64}$

Step 3.1.7

Simplify the answer.

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Step 3.1.7.1

Combine $\frac{2}{3} x^{\frac{3}{2}}]_{0}^{64}$ and $- 8 x]_{0}^{64}$ .

$\frac{2}{3} x^{\frac{3}{2}} - 8 x]_{0}^{64}$

Step 3.1.7.2

Substitute and simplify.

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Step 3.1.7.2.1

Evaluate $\frac{2}{3} x^{\frac{3}{2}} - 8 x$ at $64$ and at $0$ .

$(\frac{2}{3} \cdot 64^{\frac{3}{2}} - 8 \cdot 64) - (\frac{2}{3} \cdot 0^{\frac{3}{2}} - 8 \cdot 0)$

Step 3.1.7.2.2

Simplify.

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Step 3.1.7.2.2.1

Rewrite $64$ as $8^{2}$ .

$\frac{2}{3} \cdot {(8^{2})}^{\frac{3}{2}} - 8 \cdot 64 - (\frac{2}{3} \cdot 0^{\frac{3}{2}} - 8 \cdot 0)$

Step 3.1.7.2.2.2

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

$\frac{2}{3} \cdot 8^{2 (\frac{3}{2})} - 8 \cdot 64 - (\frac{2}{3} \cdot 0^{\frac{3}{2}} - 8 \cdot 0)$

Step 3.1.7.2.2.3

Cancel the common factor of $2$ .

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Step 3.1.7.2.2.3.1

Cancel the common factor.

$\frac{2}{3} \cdot 8^{2 (\frac{3}{2})} - 8 \cdot 64 - (\frac{2}{3} \cdot 0^{\frac{3}{2}} - 8 \cdot 0)$

Step 3.1.7.2.2.3.2

Rewrite the expression.

$\frac{2}{3} \cdot 8^{3} - 8 \cdot 64 - (\frac{2}{3} \cdot 0^{\frac{3}{2}} - 8 \cdot 0)$

Step 3.1.7.2.2.4

Raise $8$ to the power of $3$ .

$\frac{2}{3} \cdot 512 - 8 \cdot 64 - (\frac{2}{3} \cdot 0^{\frac{3}{2}} - 8 \cdot 0)$

Step 3.1.7.2.2.5

Combine $\frac{2}{3}$ and $512$ .

$\frac{2 \cdot 512}{3} - 8 \cdot 64 - (\frac{2}{3} \cdot 0^{\frac{3}{2}} - 8 \cdot 0)$

Step 3.1.7.2.2.6

Multiply $2$ by $512$ .

$\frac{1024}{3} - 8 \cdot 64 - (\frac{2}{3} \cdot 0^{\frac{3}{2}} - 8 \cdot 0)$

Step 3.1.7.2.2.7

Multiply $- 8$ by $64$ .

$\frac{1024}{3} - 512 - (\frac{2}{3} \cdot 0^{\frac{3}{2}} - 8 \cdot 0)$

Step 3.1.7.2.2.8

To write $- 512$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{1024}{3} - 512 \cdot \frac{3}{3} - (\frac{2}{3} \cdot 0^{\frac{3}{2}} - 8 \cdot 0)$

Step 3.1.7.2.2.9

Combine $- 512$ and $\frac{3}{3}$ .

$\frac{1024}{3} + \frac{- 512 \cdot 3}{3} - (\frac{2}{3} \cdot 0^{\frac{3}{2}} - 8 \cdot 0)$

Step 3.1.7.2.2.10

Combine the numerators over the common denominator.

$\frac{1024 - 512 \cdot 3}{3} - (\frac{2}{3} \cdot 0^{\frac{3}{2}} - 8 \cdot 0)$

Step 3.1.7.2.2.11

Simplify the numerator.

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Step 3.1.7.2.2.11.1

Multiply $- 512$ by $3$ .

$\frac{1024 - 1536}{3} - (\frac{2}{3} \cdot 0^{\frac{3}{2}} - 8 \cdot 0)$

Step 3.1.7.2.2.11.2

Subtract $1536$ from $1024$ .

$\frac{- 512}{3} - (\frac{2}{3} \cdot 0^{\frac{3}{2}} - 8 \cdot 0)$

Step 3.1.7.2.2.12

Move the negative in front of the fraction.

$- \frac{512}{3} - (\frac{2}{3} \cdot 0^{\frac{3}{2}} - 8 \cdot 0)$

Step 3.1.7.2.2.13

Rewrite $0$ as $0^{2}$ .

$- \frac{512}{3} - (\frac{2}{3} \cdot {(0^{2})}^{\frac{3}{2}} - 8 \cdot 0)$

Step 3.1.7.2.2.14

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

$- \frac{512}{3} - (\frac{2}{3} \cdot 0^{2 (\frac{3}{2})} - 8 \cdot 0)$

Step 3.1.7.2.2.15

Cancel the common factor of $2$ .

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Step 3.1.7.2.2.15.1

Cancel the common factor.

$- \frac{512}{3} - (\frac{2}{3} \cdot 0^{2 (\frac{3}{2})} - 8 \cdot 0)$

Step 3.1.7.2.2.15.2

Rewrite the expression.

$- \frac{512}{3} - (\frac{2}{3} \cdot 0^{3} - 8 \cdot 0)$

Step 3.1.7.2.2.16

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

$- \frac{512}{3} - (\frac{2}{3} \cdot 0 - 8 \cdot 0)$

Step 3.1.7.2.2.17

Multiply $\frac{2}{3}$ by $0$ .

$- \frac{512}{3} - (0 - 8 \cdot 0)$

Step 3.1.7.2.2.18

Multiply $- 8$ by $0$ .

$- \frac{512}{3} - (0 + 0)$

Step 3.1.7.2.2.19

Add $0$ and $0$ .

$- \frac{512}{3} - 0$

Step 3.1.7.2.2.20

Multiply $- 1$ by $0$ .

$- \frac{512}{3} + 0$

Step 3.1.7.2.2.21

Add $- \frac{512}{3}$ and $0$ .

$- \frac{512}{3}$

Step 3.2

Combine the integrals into a single integral.

$\int_{0}^{64} 8 - (\sqrt{x}) d x$

Step 3.3

Multiply $- 1$ by $\sqrt{x}$ .

$\int_{0}^{64} 8 - \sqrt{x} d x$

Step 3.4

Split the single integral into multiple integrals.

$\int_{0}^{64} 8 d x + \int_{0}^{64} - \sqrt{x} d x$

Step 3.5

Apply the constant rule.

$8 x]_{0}^{64} + \int_{0}^{64} - \sqrt{x} d x$

Step 3.6

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$8 x]_{0}^{64} - \int_{0}^{64} \sqrt{x} d x$

Step 3.7

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

$8 x]_{0}^{64} - \int_{0}^{64} x^{\frac{1}{2}} d x$

Step 3.8

By the Power Rule, the integral of $x^{\frac{1}{2}}$ with respect to $x$ is $\frac{2}{3} x^{\frac{3}{2}}$ .

$8 x]_{0}^{64} - (\frac{2}{3} x^{\frac{3}{2}}]_{0}^{64})$

Step 3.9

Simplify the answer.

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Step 3.9.1

Combine $\frac{2}{3}$ and $x^{\frac{3}{2}}$ .

$8 x]_{0}^{64} - (\frac{2 x^{\frac{3}{2}}}{3}]_{0}^{64})$

Step 3.9.2

Substitute and simplify.

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Step 3.9.2.1

Evaluate $8 x$ at $64$ and at $0$ .

$(8 \cdot 64) - 8 \cdot 0 - (\frac{2 x^{\frac{3}{2}}}{3}]_{0}^{64})$

Step 3.9.2.2

Evaluate $\frac{2 x^{\frac{3}{2}}}{3}$ at $64$ and at $0$ .

$8 \cdot 64 - 8 \cdot 0 - (\frac{2 \cdot 64^{\frac{3}{2}}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})$

Step 3.9.2.3

Simplify.

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Step 3.9.2.3.1

Multiply $8$ by $64$ .

$512 - 8 \cdot 0 - (\frac{2 \cdot 64^{\frac{3}{2}}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})$

Step 3.9.2.3.2

Multiply $- 8$ by $0$ .

$512 + 0 - (\frac{2 \cdot 64^{\frac{3}{2}}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})$

Step 3.9.2.3.3

Add $512$ and $0$ .

$512 - (\frac{2 \cdot 64^{\frac{3}{2}}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})$

Step 3.9.2.3.4

Rewrite $64$ as $8^{2}$ .

$512 - (\frac{2 \cdot {(8^{2})}^{\frac{3}{2}}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})$

Step 3.9.2.3.5

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

$512 - (\frac{2 \cdot 8^{2 (\frac{3}{2})}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})$

Step 3.9.2.3.6

Cancel the common factor of $2$ .

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Step 3.9.2.3.6.1

Cancel the common factor.

$512 - (\frac{2 \cdot 8^{2 (\frac{3}{2})}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})$

Step 3.9.2.3.6.2

Rewrite the expression.

$512 - (\frac{2 \cdot 8^{3}}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})$

Step 3.9.2.3.7

Raise $8$ to the power of $3$ .

$512 - (\frac{2 \cdot 512}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})$

Step 3.9.2.3.8

Multiply $2$ by $512$ .

$512 - (\frac{1024}{3} - \frac{2 \cdot 0^{\frac{3}{2}}}{3})$

Step 3.9.2.3.9

Rewrite $0$ as $0^{2}$ .

$512 - (\frac{1024}{3} - \frac{2 \cdot {(0^{2})}^{\frac{3}{2}}}{3})$

Step 3.9.2.3.10

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

$512 - (\frac{1024}{3} - \frac{2 \cdot 0^{2 (\frac{3}{2})}}{3})$

Step 3.9.2.3.11

Cancel the common factor of $2$ .

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Step 3.9.2.3.11.1

Cancel the common factor.

$512 - (\frac{1024}{3} - \frac{2 \cdot 0^{2 (\frac{3}{2})}}{3})$

Step 3.9.2.3.11.2

Rewrite the expression.

$512 - (\frac{1024}{3} - \frac{2 \cdot 0^{3}}{3})$

Step 3.9.2.3.12

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

$512 - (\frac{1024}{3} - \frac{2 \cdot 0}{3})$

Step 3.9.2.3.13

Multiply $2$ by $0$ .

$512 - (\frac{1024}{3} - \frac{0}{3})$

Step 3.9.2.3.14

Cancel the common factor of $0$ and $3$ .

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Step 3.9.2.3.14.1

Factor $3$ out of $0$ .

$512 - (\frac{1024}{3} - \frac{3 (0)}{3})$

Step 3.9.2.3.14.2

Cancel the common factors.

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Step 3.9.2.3.14.2.1

Factor $3$ out of $3$ .

$512 - (\frac{1024}{3} - \frac{3 \cdot 0}{3 \cdot 1})$

Step 3.9.2.3.14.2.2

Cancel the common factor.

$512 - (\frac{1024}{3} - \frac{3 \cdot 0}{3 \cdot 1})$

Step 3.9.2.3.14.2.3

Rewrite the expression.

$512 - (\frac{1024}{3} - \frac{0}{1})$

Step 3.9.2.3.14.2.4

Divide $0$ by $1$ .

$512 - (\frac{1024}{3} - 0)$

Step 3.9.2.3.15

Multiply $- 1$ by $0$ .

$512 - (\frac{1024}{3} + 0)$

Step 3.9.2.3.16

Add $\frac{1024}{3}$ and $0$ .

$512 - \frac{1024}{3}$

Step 3.9.2.3.17

To write $512$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$512 \cdot \frac{3}{3} - \frac{1024}{3}$

Step 3.9.2.3.18

Combine $512$ and $\frac{3}{3}$ .

$\frac{512 \cdot 3}{3} - \frac{1024}{3}$

Step 3.9.2.3.19

Combine the numerators over the common denominator.

$\frac{512 \cdot 3 - 1024}{3}$

Step 3.9.2.3.20

Simplify the numerator.

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Step 3.9.2.3.20.1

Multiply $512$ by $3$ .

$\frac{1536 - 1024}{3}$

Step 3.9.2.3.20.2

Subtract $1024$ from $1536$ .

$\frac{512}{3}$

Step 4