Calculus Examples

$y = x^{\frac{8}{7}}$ , $y = 6 x^{\frac{1}{7}}$

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.

$x^{\frac{8}{7}} = 6 x^{\frac{1}{7}}$

Step 1.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = 0, 6$

Step 1.3

Evaluate $y$ when $x = 0$ .

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Step 1.3.1

Substitute $0$ for $x$ .

$y = 6 {(0)}^{\frac{1}{7}}$

Step 1.3.2

Substitute $0$ for $x$ in $y = 6 {(0)}^{\frac{1}{7}}$ and solve for $y$ .

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Step 1.3.2.1

Remove parentheses.

$y = 6 \cdot 0^{\frac{1}{7}}$

Step 1.3.2.2

Remove parentheses.

$y = 6 {(0)}^{\frac{1}{7}}$

Step 1.3.2.3

Simplify $6 {(0)}^{\frac{1}{7}}$ .

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Step 1.3.2.3.1

Simplify the expression.

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Step 1.3.2.3.1.1

Rewrite $0$ as $0^{7}$ .

$y = 6 {(0^{7})}^{\frac{1}{7}}$

Step 1.3.2.3.1.2

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

$y = 6 \cdot 0^{7 (\frac{1}{7})}$

Step 1.3.2.3.2

Cancel the common factor of $7$ .

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Step 1.3.2.3.2.1

Cancel the common factor.

$y = 6 \cdot 0^{7 (\frac{1}{7})}$

Step 1.3.2.3.2.2

Rewrite the expression.

$y = 6 \cdot 0^{1}$

Step 1.3.2.3.3

Evaluate the exponent.

$y = 6 \cdot 0$

Step 1.3.2.3.4

Multiply $6$ by $0$ .

$y = 0$

Step 1.4

Evaluate $y$ when $x = 6$ .

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Step 1.4.1

Substitute $6$ for $x$ .

$y = 6 {(6)}^{\frac{1}{7}}$

Step 1.4.2

Substitute $6$ for $x$ in $y = 6 {(6)}^{\frac{1}{7}}$ and solve for $y$ .

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Step 1.4.2.1

Remove parentheses.

$y = 6 \cdot 6^{\frac{1}{7}}$

Step 1.4.2.2

Remove parentheses.

$y = 6 {(6)}^{\frac{1}{7}}$

Step 1.4.2.3

Multiply $6$ by ${(6)}^{\frac{1}{7}}$ by adding the exponents.

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Step 1.4.2.3.1

Multiply $6$ by ${(6)}^{\frac{1}{7}}$ .

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Step 1.4.2.3.1.1

Raise $6$ to the power of $1$ .

$y = 6^{1} {(6)}^{\frac{1}{7}}$

Step 1.4.2.3.1.2

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

$y = 6^{1 + \frac{1}{7}}$

Step 1.4.2.3.2

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

$y = 6^{\frac{7}{7} + \frac{1}{7}}$

Step 1.4.2.3.3

Combine the numerators over the common denominator.

$y = 6^{\frac{7 + 1}{7}}$

Step 1.4.2.3.4

Add $7$ and $1$ .

$y = 6^{\frac{8}{7}}$

Step 1.5

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

$(0, 0)$

$(6, 6^{\frac{8}{7}})$

$(0, 0)$

$(6, 6^{\frac{8}{7}})$

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.

$A r e a = \int_{0}^{6} 6 x^{\frac{1}{7}} d x - \int_{0}^{6} x^{\frac{8}{7}} d x$

Step 3

Integrate to find the area between $0$ and $6$ .

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Step 3.1

Combine the integrals into a single integral.

$\int_{0}^{6} 6 x^{\frac{1}{7}} - (x^{\frac{8}{7}}) d x$

Step 3.2

Multiply $- 1$ by $x^{\frac{8}{7}}$ .

$\int_{0}^{6} 6 x^{\frac{1}{7}} - x^{\frac{8}{7}} d x$

Step 3.3

Split the single integral into multiple integrals.

$\int_{0}^{6} 6 x^{\frac{1}{7}} d x + \int_{0}^{6} - x^{\frac{8}{7}} d x$

Step 3.4

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$6 \int_{0}^{6} x^{\frac{1}{7}} d x + \int_{0}^{6} - x^{\frac{8}{7}} d x$

Step 3.5

By the Power Rule, the integral of $x^{\frac{1}{7}}$ with respect to $x$ is $\frac{7}{8} x^{\frac{8}{7}}$ .

$6 (\frac{7}{8} x^{\frac{8}{7}}]_{0}^{6}) + \int_{0}^{6} - x^{\frac{8}{7}} d x$

Step 3.6

Combine $\frac{7}{8}$ and $x^{\frac{8}{7}}$ .

$6 (\frac{7 x^{\frac{8}{7}}}{8}]_{0}^{6}) + \int_{0}^{6} - x^{\frac{8}{7}} d x$

Step 3.7

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

$6 (\frac{7 x^{\frac{8}{7}}}{8}]_{0}^{6}) - \int_{0}^{6} x^{\frac{8}{7}} d x$

Step 3.8

By the Power Rule, the integral of $x^{\frac{8}{7}}$ with respect to $x$ is $\frac{7}{15} x^{\frac{15}{7}}$ .

$6 (\frac{7 x^{\frac{8}{7}}}{8}]_{0}^{6}) - (\frac{7}{15} x^{\frac{15}{7}}]_{0}^{6})$

Step 3.9

Simplify the answer.

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Step 3.9.1

Combine $\frac{7}{15}$ and $x^{\frac{15}{7}}$ .

$6 (\frac{7 x^{\frac{8}{7}}}{8}]_{0}^{6}) - (\frac{7 x^{\frac{15}{7}}}{15}]_{0}^{6})$

Step 3.9.2

Substitute and simplify.

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Step 3.9.2.1

Evaluate $\frac{7 x^{\frac{8}{7}}}{8}$ at $6$ and at $0$ .

$6 ((\frac{7 \cdot 6^{\frac{8}{7}}}{8}) - \frac{7 \cdot 0^{\frac{8}{7}}}{8}) - (\frac{7 x^{\frac{15}{7}}}{15}]_{0}^{6})$

Step 3.9.2.2

Evaluate $\frac{7 x^{\frac{15}{7}}}{15}$ at $6$ and at $0$ .

$6 (\frac{7 \cdot 6^{\frac{8}{7}}}{8} - \frac{7 \cdot 0^{\frac{8}{7}}}{8}) - (\frac{7 \cdot 6^{\frac{15}{7}}}{15} - \frac{7 \cdot 0^{\frac{15}{7}}}{15})$

Step 3.9.2.3

Simplify.

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Step 3.9.2.3.1

Rewrite $0$ as $0^{7}$ .

$6 (\frac{7 \cdot 6^{\frac{8}{7}}}{8} - \frac{7 \cdot {(0^{7})}^{\frac{8}{7}}}{8}) - (\frac{7 \cdot 6^{\frac{15}{7}}}{15} - \frac{7 \cdot 0^{\frac{15}{7}}}{15})$

Step 3.9.2.3.2

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

$6 (\frac{7 \cdot 6^{\frac{8}{7}}}{8} - \frac{7 \cdot 0^{7 (\frac{8}{7})}}{8}) - (\frac{7 \cdot 6^{\frac{15}{7}}}{15} - \frac{7 \cdot 0^{\frac{15}{7}}}{15})$

Step 3.9.2.3.3

Cancel the common factor of $7$ .

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Step 3.9.2.3.3.1

Cancel the common factor.

$6 (\frac{7 \cdot 6^{\frac{8}{7}}}{8} - \frac{7 \cdot 0^{7 (\frac{8}{7})}}{8}) - (\frac{7 \cdot 6^{\frac{15}{7}}}{15} - \frac{7 \cdot 0^{\frac{15}{7}}}{15})$

Step 3.9.2.3.3.2

Rewrite the expression.

$6 (\frac{7 \cdot 6^{\frac{8}{7}}}{8} - \frac{7 \cdot 0^{8}}{8}) - (\frac{7 \cdot 6^{\frac{15}{7}}}{15} - \frac{7 \cdot 0^{\frac{15}{7}}}{15})$

Step 3.9.2.3.4

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

$6 (\frac{7 \cdot 6^{\frac{8}{7}}}{8} - \frac{7 \cdot 0}{8}) - (\frac{7 \cdot 6^{\frac{15}{7}}}{15} - \frac{7 \cdot 0^{\frac{15}{7}}}{15})$

Step 3.9.2.3.5

Multiply $7$ by $0$ .

$6 (\frac{7 \cdot 6^{\frac{8}{7}}}{8} - \frac{0}{8}) - (\frac{7 \cdot 6^{\frac{15}{7}}}{15} - \frac{7 \cdot 0^{\frac{15}{7}}}{15})$

Step 3.9.2.3.6

Cancel the common factor of $0$ and $8$ .

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Step 3.9.2.3.6.1

Factor $8$ out of $0$ .

$6 (\frac{7 \cdot 6^{\frac{8}{7}}}{8} - \frac{8 (0)}{8}) - (\frac{7 \cdot 6^{\frac{15}{7}}}{15} - \frac{7 \cdot 0^{\frac{15}{7}}}{15})$

Step 3.9.2.3.6.2

Cancel the common factors.

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Step 3.9.2.3.6.2.1

Factor $8$ out of $8$ .

$6 (\frac{7 \cdot 6^{\frac{8}{7}}}{8} - \frac{8 \cdot 0}{8 \cdot 1}) - (\frac{7 \cdot 6^{\frac{15}{7}}}{15} - \frac{7 \cdot 0^{\frac{15}{7}}}{15})$

Step 3.9.2.3.6.2.2

Cancel the common factor.

$6 (\frac{7 \cdot 6^{\frac{8}{7}}}{8} - \frac{8 \cdot 0}{8 \cdot 1}) - (\frac{7 \cdot 6^{\frac{15}{7}}}{15} - \frac{7 \cdot 0^{\frac{15}{7}}}{15})$

Step 3.9.2.3.6.2.3

Rewrite the expression.

$6 (\frac{7 \cdot 6^{\frac{8}{7}}}{8} - \frac{0}{1}) - (\frac{7 \cdot 6^{\frac{15}{7}}}{15} - \frac{7 \cdot 0^{\frac{15}{7}}}{15})$

Step 3.9.2.3.6.2.4

Divide $0$ by $1$ .

$6 (\frac{7 \cdot 6^{\frac{8}{7}}}{8} - 0) - (\frac{7 \cdot 6^{\frac{15}{7}}}{15} - \frac{7 \cdot 0^{\frac{15}{7}}}{15})$

Step 3.9.2.3.7

Multiply $- 1$ by $0$ .

$6 (\frac{7 \cdot 6^{\frac{8}{7}}}{8} + 0) - (\frac{7 \cdot 6^{\frac{15}{7}}}{15} - \frac{7 \cdot 0^{\frac{15}{7}}}{15})$

Step 3.9.2.3.8

Add $\frac{7 \cdot 6^{\frac{8}{7}}}{8}$ and $0$ .

$6 \frac{7 \cdot 6^{\frac{8}{7}}}{8} - (\frac{7 \cdot 6^{\frac{15}{7}}}{15} - \frac{7 \cdot 0^{\frac{15}{7}}}{15})$

Step 3.9.2.3.9

Combine $6$ and $\frac{7 \cdot 6^{\frac{8}{7}}}{8}$ .

$\frac{6 (7 \cdot 6^{\frac{8}{7}})}{8} - (\frac{7 \cdot 6^{\frac{15}{7}}}{15} - \frac{7 \cdot 0^{\frac{15}{7}}}{15})$

Step 3.9.2.3.10

Multiply $7$ by $6$ .

$\frac{42 \cdot 6^{\frac{8}{7}}}{8} - (\frac{7 \cdot 6^{\frac{15}{7}}}{15} - \frac{7 \cdot 0^{\frac{15}{7}}}{15})$

Step 3.9.2.3.11

Factor $2$ out of $42 \cdot 6^{\frac{8}{7}}$ .

$\frac{2 (21 \cdot 6^{\frac{8}{7}})}{8} - (\frac{7 \cdot 6^{\frac{15}{7}}}{15} - \frac{7 \cdot 0^{\frac{15}{7}}}{15})$

Step 3.9.2.3.12

Cancel the common factors.

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Step 3.9.2.3.12.1

Factor $2$ out of $8$ .

$\frac{2 (21 \cdot 6^{\frac{8}{7}})}{2 (4)} - (\frac{7 \cdot 6^{\frac{15}{7}}}{15} - \frac{7 \cdot 0^{\frac{15}{7}}}{15})$

Step 3.9.2.3.12.2

Cancel the common factor.

$\frac{2 (21 \cdot 6^{\frac{8}{7}})}{2 \cdot 4} - (\frac{7 \cdot 6^{\frac{15}{7}}}{15} - \frac{7 \cdot 0^{\frac{15}{7}}}{15})$

Step 3.9.2.3.12.3

Rewrite the expression.

$\frac{21 \cdot 6^{\frac{8}{7}}}{4} - (\frac{7 \cdot 6^{\frac{15}{7}}}{15} - \frac{7 \cdot 0^{\frac{15}{7}}}{15})$

Step 3.9.2.3.13

Rewrite $0$ as $0^{7}$ .

$\frac{21 \cdot 6^{\frac{8}{7}}}{4} - (\frac{7 \cdot 6^{\frac{15}{7}}}{15} - \frac{7 \cdot {(0^{7})}^{\frac{15}{7}}}{15})$

Step 3.9.2.3.14

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

$\frac{21 \cdot 6^{\frac{8}{7}}}{4} - (\frac{7 \cdot 6^{\frac{15}{7}}}{15} - \frac{7 \cdot 0^{7 (\frac{15}{7})}}{15})$

Step 3.9.2.3.15

Cancel the common factor of $7$ .

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Step 3.9.2.3.15.1

Cancel the common factor.

$\frac{21 \cdot 6^{\frac{8}{7}}}{4} - (\frac{7 \cdot 6^{\frac{15}{7}}}{15} - \frac{7 \cdot 0^{7 (\frac{15}{7})}}{15})$

Step 3.9.2.3.15.2

Rewrite the expression.

$\frac{21 \cdot 6^{\frac{8}{7}}}{4} - (\frac{7 \cdot 6^{\frac{15}{7}}}{15} - \frac{7 \cdot 0^{15}}{15})$

Step 3.9.2.3.16

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

$\frac{21 \cdot 6^{\frac{8}{7}}}{4} - (\frac{7 \cdot 6^{\frac{15}{7}}}{15} - \frac{7 \cdot 0}{15})$

Step 3.9.2.3.17

Multiply $7$ by $0$ .

$\frac{21 \cdot 6^{\frac{8}{7}}}{4} - (\frac{7 \cdot 6^{\frac{15}{7}}}{15} - \frac{0}{15})$

Step 3.9.2.3.18

Cancel the common factor of $0$ and $15$ .

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Step 3.9.2.3.18.1

Factor $15$ out of $0$ .

$\frac{21 \cdot 6^{\frac{8}{7}}}{4} - (\frac{7 \cdot 6^{\frac{15}{7}}}{15} - \frac{15 (0)}{15})$

Step 3.9.2.3.18.2

Cancel the common factors.

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Step 3.9.2.3.18.2.1

Factor $15$ out of $15$ .

$\frac{21 \cdot 6^{\frac{8}{7}}}{4} - (\frac{7 \cdot 6^{\frac{15}{7}}}{15} - \frac{15 \cdot 0}{15 \cdot 1})$

Step 3.9.2.3.18.2.2

Cancel the common factor.

$\frac{21 \cdot 6^{\frac{8}{7}}}{4} - (\frac{7 \cdot 6^{\frac{15}{7}}}{15} - \frac{15 \cdot 0}{15 \cdot 1})$

Step 3.9.2.3.18.2.3

Rewrite the expression.

$\frac{21 \cdot 6^{\frac{8}{7}}}{4} - (\frac{7 \cdot 6^{\frac{15}{7}}}{15} - \frac{0}{1})$

Step 3.9.2.3.18.2.4

Divide $0$ by $1$ .

$\frac{21 \cdot 6^{\frac{8}{7}}}{4} - (\frac{7 \cdot 6^{\frac{15}{7}}}{15} - 0)$

Step 3.9.2.3.19

Multiply $- 1$ by $0$ .

$\frac{21 \cdot 6^{\frac{8}{7}}}{4} - (\frac{7 \cdot 6^{\frac{15}{7}}}{15} + 0)$

Step 3.9.2.3.20

Add $\frac{7 \cdot 6^{\frac{15}{7}}}{15}$ and $0$ .

$\frac{21 \cdot 6^{\frac{8}{7}}}{4} - \frac{7 \cdot 6^{\frac{15}{7}}}{15}$

Step 3.9.2.3.21

To write $\frac{21 \cdot 6^{\frac{8}{7}}}{4}$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$\frac{21 \cdot 6^{\frac{8}{7}}}{4} \cdot \frac{15}{15} - \frac{7 \cdot 6^{\frac{15}{7}}}{15}$

Step 3.9.2.3.22

To write $- \frac{7 \cdot 6^{\frac{15}{7}}}{15}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{21 \cdot 6^{\frac{8}{7}}}{4} \cdot \frac{15}{15} - \frac{7 \cdot 6^{\frac{15}{7}}}{15} \cdot \frac{4}{4}$

Step 3.9.2.3.23

Write each expression with a common denominator of $60$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.9.2.3.23.1

Multiply $\frac{21 \cdot 6^{\frac{8}{7}}}{4}$ by $\frac{15}{15}$ .

$\frac{21 \cdot 6^{\frac{8}{7}} \cdot 15}{4 \cdot 15} - \frac{7 \cdot 6^{\frac{15}{7}}}{15} \cdot \frac{4}{4}$

Step 3.9.2.3.23.2

Multiply $4$ by $15$ .

$\frac{21 \cdot 6^{\frac{8}{7}} \cdot 15}{60} - \frac{7 \cdot 6^{\frac{15}{7}}}{15} \cdot \frac{4}{4}$

Step 3.9.2.3.23.3

Multiply $\frac{7 \cdot 6^{\frac{15}{7}}}{15}$ by $\frac{4}{4}$ .

$\frac{21 \cdot 6^{\frac{8}{7}} \cdot 15}{60} - \frac{7 \cdot 6^{\frac{15}{7}} \cdot 4}{15 \cdot 4}$

Step 3.9.2.3.23.4

Multiply $15$ by $4$ .

$\frac{21 \cdot 6^{\frac{8}{7}} \cdot 15}{60} - \frac{7 \cdot 6^{\frac{15}{7}} \cdot 4}{60}$

Step 3.9.2.3.24

Combine the numerators over the common denominator.

$\frac{21 \cdot 6^{\frac{8}{7}} \cdot 15 - 7 \cdot 6^{\frac{15}{7}} \cdot 4}{60}$

Step 3.9.2.3.25

Multiply $15$ by $21$ .

$\frac{315 \cdot 6^{\frac{8}{7}} - 7 \cdot 6^{\frac{15}{7}} \cdot 4}{60}$

Step 3.9.2.3.26

Multiply $4$ by $- 7$ .

$\frac{315 \cdot 6^{\frac{8}{7}} - 28 \cdot 6^{\frac{15}{7}}}{60}$

Step 4