Calculus Examples

$y = x^{4} - 4 x^{2} + 1$ , $y = x^{2} - 3$

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.

$x^{4} - 4 x^{2} + 1 = x^{2} - 3$

Step 1.2

Solve $x^{4} - 4 x^{2} + 1 = x^{2} - 3$ for $x$ .

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Step 1.2.1

Move all terms containing $x$ to the left side of the equation.

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Step 1.2.1.1

Subtract $x^{2}$ from both sides of the equation.

$x^{4} - 4 x^{2} + 1 - x^{2} = - 3$

Step 1.2.1.2

Subtract $x^{2}$ from $- 4 x^{2}$ .

$x^{4} - 5 x^{2} + 1 = - 3$

Step 1.2.2

Substitute $u = x^{2}$ into the equation. This will make the quadratic formula easy to use.

$u^{2} - 5 u + 1 = - 3$

$u = x^{2} + y = x^{2} - 3$

Step 1.2.3

Add $3$ to both sides of the equation.

$u^{2} - 5 u + 1 + 3 = 0$

Step 1.2.4

Add $1$ and $3$ .

$u^{2} - 5 u + 4 = 0$

Step 1.2.5

Factor $u^{2} - 5 u + 4$ using the AC method.

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Step 1.2.5.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $4$ and whose sum is $- 5$ .

$- 4, - 1 + y = x^{2} - 3$

Step 1.2.5.2

Write the factored form using these integers.

$(u - 4) (u - 1) = 0$

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 $0$ .

$u - 4 = 0$

$u - 1 = 0 + y = x^{2} - 3$

Step 1.2.7

Set $u - 4$ equal to $0$ and solve for $u$ .

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Step 1.2.7.1

Set $u - 4$ equal to $0$ .

$u - 4 = 0$

Step 1.2.7.2

Add $4$ to both sides of the equation.

$u = 4$

Step 1.2.8

Set $u - 1$ equal to $0$ and solve for $u$ .

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Step 1.2.8.1

Set $u - 1$ equal to $0$ .

$u - 1 = 0$

Step 1.2.8.2

Add $1$ to both sides of the equation.

$u = 1$

Step 1.2.9

The final solution is all the values that make $(u - 4) (u - 1) = 0$ true.

$u = 4, 1$

Step 1.2.10

Substitute the real value of $u = x^{2}$ back into the solved equation.

$x^{2} = 4$

$x^{2} = 1 + y = x^{2} - 3$

Step 1.2.11

Solve the first equation for $x$ .

$x^{2} = 4$

Step 1.2.12

Solve the equation for $x$ .

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Step 1.2.12.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{4}$

Step 1.2.12.2

Simplify $\pm \sqrt{4}$ .

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Step 1.2.12.2.1

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

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

Step 1.2.12.2.2

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

$x = \pm 2$

Step 1.2.12.3

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.12.3.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 2$

Step 1.2.12.3.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 2$

Step 1.2.12.3.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 2, - 2$

Step 1.2.13

Solve the second equation for $x$ .

${(x^{2})}^{1} = 1$

Step 1.2.14

Solve the equation for $x$ .

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Step 1.2.14.1

Remove parentheses.

$x^{2} = 1$

Step 1.2.14.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{1}$

Step 1.2.14.3

Any root of $1$ is $1$ .

$x = \pm 1$

Step 1.2.14.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.14.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 1$

Step 1.2.14.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 1$

Step 1.2.14.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 1, - 1$

Step 1.2.15

The solution to $x^{4} - 5 x^{2} + 1 = - 3$ is $x = 2, - 2, 1, - 1$ .

$x = 2, - 2, 1, - 1$

Step 1.3

Evaluate $y$ when $x = 2$ .

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Step 1.3.1

Substitute $2$ for $x$ .

$y = {(2)}^{2} - 3$

Step 1.3.2

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

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Step 1.3.2.1

Remove parentheses.

$y = 2^{2} - 3$

Step 1.3.2.2

Simplify $2^{2} - 3$ .

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Step 1.3.2.2.1

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

$y = 4 - 3$

Step 1.3.2.2.2

Subtract $3$ from $4$ .

$y = 1$

Step 1.4

Evaluate $y$ when $x = 1$ .

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Step 1.4.1

Substitute $1$ for $x$ .

$y = {(1)}^{2} - 3$

Step 1.4.2

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

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Step 1.4.2.1

Remove parentheses.

$y = 1^{2} - 3$

Step 1.4.2.2

Simplify $1^{2} - 3$ .

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Step 1.4.2.2.1

One to any power is one.

$y = 1 - 3$

Step 1.4.2.2.2

Subtract $3$ from $1$ .

$y = - 2$

Step 1.5

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

$(2, 1)$

$(- 2, 1)$

$(1, - 2)$

$(- 1, - 2)$

$(2, 1)$

$(- 2, 1)$

$(1, - 2)$

$(- 1, - 2)$

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.

$A r e a = \int_{- 2}^{- 1} x^{2} - 3 d x - \int_{- 2}^{- 1} x^{4} - 4 x^{2} + 1 d x$

Step 3

Integrate to find the area between $- 2$ and $- 1$ .

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Step 3.1

Combine the integrals into a single integral.

$\int_{- 2}^{- 1} x^{2} - 3 - (x^{4} - 4 x^{2} + 1) d x$

Step 3.2

Simplify each term.

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Step 3.2.1

Apply the distributive property.

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

Step 3.2.2

Simplify.

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Step 3.2.2.1

Multiply $- 4$ by $- 1$ .

$x^{2} - 3 - x^{4} + 4 x^{2} - 1 \cdot 1$

Step 3.2.2.2

Multiply $- 1$ by $1$ .

$x^{2} - 3 - x^{4} + 4 x^{2} - 1$

$\int_{- 2}^{- 1} x^{2} - 3 - x^{4} + 4 x^{2} - 1 d x$

Step 3.3

Simplify by adding terms.

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Step 3.3.1

Add $x^{2}$ and $4 x^{2}$ .

$5 x^{2} - 3 - x^{4} - 1$

Step 3.3.2

Subtract $1$ from $- 3$ .

$5 x^{2} - x^{4} - 4$

$\int_{- 2}^{- 1} 5 x^{2} - x^{4} - 4 d x$

Step 3.4

Split the single integral into multiple integrals.

$\int_{- 2}^{- 1} 5 x^{2} d x + \int_{- 2}^{- 1} - x^{4} d x + \int_{- 2}^{- 1} - 4 d x$

Step 3.5

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int_{- 2}^{- 1} x^{2} d x + \int_{- 2}^{- 1} - x^{4} d x + \int_{- 2}^{- 1} - 4 d x$

Step 3.6

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

$5 (\frac{1}{3} x^{3}]_{- 2}^{- 1}) + \int_{- 2}^{- 1} - x^{4} d x + \int_{- 2}^{- 1} - 4 d x$

Step 3.7

Combine $\frac{1}{3}$ and $x^{3}$ .

$5 (\frac{x^{3}}{3}]_{- 2}^{- 1}) + \int_{- 2}^{- 1} - x^{4} d x + \int_{- 2}^{- 1} - 4 d x$

Step 3.8

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

$5 (\frac{x^{3}}{3}]_{- 2}^{- 1}) - \int_{- 2}^{- 1} x^{4} d x + \int_{- 2}^{- 1} - 4 d x$

Step 3.9

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$5 (\frac{x^{3}}{3}]_{- 2}^{- 1}) - (\frac{1}{5} x^{5}]_{- 2}^{- 1}) + \int_{- 2}^{- 1} - 4 d x$

Step 3.10

Combine $\frac{1}{5}$ and $x^{5}$ .

$5 (\frac{x^{3}}{3}]_{- 2}^{- 1}) - (\frac{x^{5}}{5}]_{- 2}^{- 1}) + \int_{- 2}^{- 1} - 4 d x$

Step 3.11

Apply the constant rule.

$5 (\frac{x^{3}}{3}]_{- 2}^{- 1}) - (\frac{x^{5}}{5}]_{- 2}^{- 1}) + - 4 x]_{- 2}^{- 1}$

Step 3.12

Substitute and simplify.

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Step 3.12.1

Evaluate $\frac{x^{3}}{3}$ at $- 1$ and at $- 2$ .

$5 ((\frac{{(- 1)}^{3}}{3}) - \frac{{(- 2)}^{3}}{3}) - (\frac{x^{5}}{5}]_{- 2}^{- 1}) + - 4 x]_{- 2}^{- 1}$

Step 3.12.2

Evaluate $\frac{x^{5}}{5}$ at $- 1$ and at $- 2$ .

$5 (\frac{{(- 1)}^{3}}{3} - \frac{{(- 2)}^{3}}{3}) - (\frac{{(- 1)}^{5}}{5} - \frac{{(- 2)}^{5}}{5}) + - 4 x]_{- 2}^{- 1}$

Step 3.12.3

Evaluate $- 4 x$ at $- 1$ and at $- 2$ .

$5 (\frac{{(- 1)}^{3}}{3} - \frac{{(- 2)}^{3}}{3}) - (\frac{{(- 1)}^{5}}{5} - \frac{{(- 2)}^{5}}{5}) + (- 4 \cdot - 1) + 4 \cdot - 2$

Step 3.12.4

Simplify.

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Step 3.12.4.1

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

$5 (\frac{- 1}{3} - \frac{{(- 2)}^{3}}{3}) - (\frac{{(- 1)}^{5}}{5} - \frac{{(- 2)}^{5}}{5}) + (- 4 \cdot - 1) + 4 \cdot - 2$

Step 3.12.4.2

Move the negative in front of the fraction.

$5 (- \frac{1}{3} - \frac{{(- 2)}^{3}}{3}) - (\frac{{(- 1)}^{5}}{5} - \frac{{(- 2)}^{5}}{5}) + (- 4 \cdot - 1) + 4 \cdot - 2$

Step 3.12.4.3

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

$5 (- \frac{1}{3} - \frac{- 8}{3}) - (\frac{{(- 1)}^{5}}{5} - \frac{{(- 2)}^{5}}{5}) + (- 4 \cdot - 1) + 4 \cdot - 2$

Step 3.12.4.4

Move the negative in front of the fraction.

$5 (- \frac{1}{3} - - \frac{8}{3}) - (\frac{{(- 1)}^{5}}{5} - \frac{{(- 2)}^{5}}{5}) + (- 4 \cdot - 1) + 4 \cdot - 2$

Step 3.12.4.5

Multiply $- 1$ by $- 1$ .

$5 (- \frac{1}{3} + 1 (\frac{8}{3})) - (\frac{{(- 1)}^{5}}{5} - \frac{{(- 2)}^{5}}{5}) + (- 4 \cdot - 1) + 4 \cdot - 2$

Step 3.12.4.6

Multiply $\frac{8}{3}$ by $1$ .

$5 (- \frac{1}{3} + \frac{8}{3}) - (\frac{{(- 1)}^{5}}{5} - \frac{{(- 2)}^{5}}{5}) + (- 4 \cdot - 1) + 4 \cdot - 2$

Step 3.12.4.7

Combine the numerators over the common denominator.

$5 \frac{- 1 + 8}{3} - (\frac{{(- 1)}^{5}}{5} - \frac{{(- 2)}^{5}}{5}) + (- 4 \cdot - 1) + 4 \cdot - 2$

Step 3.12.4.8

Add $- 1$ and $8$ .

$5 (\frac{7}{3}) - (\frac{{(- 1)}^{5}}{5} - \frac{{(- 2)}^{5}}{5}) + (- 4 \cdot - 1) + 4 \cdot - 2$

Step 3.12.4.9

Combine $5$ and $\frac{7}{3}$ .

$\frac{5 \cdot 7}{3} - (\frac{{(- 1)}^{5}}{5} - \frac{{(- 2)}^{5}}{5}) + (- 4 \cdot - 1) + 4 \cdot - 2$

Step 3.12.4.10

Multiply $5$ by $7$ .

$\frac{35}{3} - (\frac{{(- 1)}^{5}}{5} - \frac{{(- 2)}^{5}}{5}) + (- 4 \cdot - 1) + 4 \cdot - 2$

Step 3.12.4.11

Raise $- 1$ to the power of $5$ .

$\frac{35}{3} - (\frac{- 1}{5} - \frac{{(- 2)}^{5}}{5}) + (- 4 \cdot - 1) + 4 \cdot - 2$

Step 3.12.4.12

Move the negative in front of the fraction.

$\frac{35}{3} - (- \frac{1}{5} - \frac{{(- 2)}^{5}}{5}) + (- 4 \cdot - 1) + 4 \cdot - 2$

Step 3.12.4.13

Raise $- 2$ to the power of $5$ .

$\frac{35}{3} - (- \frac{1}{5} - \frac{- 32}{5}) + (- 4 \cdot - 1) + 4 \cdot - 2$

Step 3.12.4.14

Move the negative in front of the fraction.

$\frac{35}{3} - (- \frac{1}{5} - - \frac{32}{5}) + (- 4 \cdot - 1) + 4 \cdot - 2$

Step 3.12.4.15

Multiply $- 1$ by $- 1$ .

$\frac{35}{3} - (- \frac{1}{5} + 1 (\frac{32}{5})) + (- 4 \cdot - 1) + 4 \cdot - 2$

Step 3.12.4.16

Multiply $\frac{32}{5}$ by $1$ .

$\frac{35}{3} - (- \frac{1}{5} + \frac{32}{5}) + (- 4 \cdot - 1) + 4 \cdot - 2$

Step 3.12.4.17

Combine the numerators over the common denominator.

$\frac{35}{3} - \frac{- 1 + 32}{5} + (- 4 \cdot - 1) + 4 \cdot - 2$

Step 3.12.4.18

Add $- 1$ and $32$ .

$\frac{35}{3} - \frac{31}{5} + (- 4 \cdot - 1) + 4 \cdot - 2$

Step 3.12.4.19

To write $\frac{35}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{35}{3} \cdot \frac{5}{5} - \frac{31}{5} + (- 4 \cdot - 1) + 4 \cdot - 2$

Step 3.12.4.20

To write $- \frac{31}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{35}{3} \cdot \frac{5}{5} - \frac{31}{5} \cdot \frac{3}{3} + (- 4 \cdot - 1) + 4 \cdot - 2$

Step 3.12.4.21

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 3.12.4.21.1

Multiply $\frac{35}{3}$ by $\frac{5}{5}$ .

$\frac{35 \cdot 5}{3 \cdot 5} - \frac{31}{5} \cdot \frac{3}{3} + (- 4 \cdot - 1) + 4 \cdot - 2$

Step 3.12.4.21.2

Multiply $3$ by $5$ .

$\frac{35 \cdot 5}{15} - \frac{31}{5} \cdot \frac{3}{3} + (- 4 \cdot - 1) + 4 \cdot - 2$

Step 3.12.4.21.3

Multiply $\frac{31}{5}$ by $\frac{3}{3}$ .

$\frac{35 \cdot 5}{15} - \frac{31 \cdot 3}{5 \cdot 3} + (- 4 \cdot - 1) + 4 \cdot - 2$

Step 3.12.4.21.4

Multiply $5$ by $3$ .

$\frac{35 \cdot 5}{15} - \frac{31 \cdot 3}{15} + (- 4 \cdot - 1) + 4 \cdot - 2$

Step 3.12.4.22

Combine the numerators over the common denominator.

$\frac{35 \cdot 5 - 31 \cdot 3}{15} + (- 4 \cdot - 1) + 4 \cdot - 2$

Step 3.12.4.23

Simplify the numerator.

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Step 3.12.4.23.1

Multiply $35$ by $5$ .

$\frac{175 - 31 \cdot 3}{15} + (- 4 \cdot - 1) + 4 \cdot - 2$

Step 3.12.4.23.2

Multiply $- 31$ by $3$ .

$\frac{175 - 93}{15} + (- 4 \cdot - 1) + 4 \cdot - 2$

Step 3.12.4.23.3

Subtract $93$ from $175$ .

$\frac{82}{15} + (- 4 \cdot - 1) + 4 \cdot - 2$

Step 3.12.4.24

Multiply $- 4$ by $- 1$ .

$\frac{82}{15} + 4 + 4 \cdot - 2$

Step 3.12.4.25

Multiply $4$ by $- 2$ .

$\frac{82}{15} + 4 - 8$

Step 3.12.4.26

Subtract $8$ from $4$ .

$\frac{82}{15} - 4$

Step 3.12.4.27

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

$\frac{82}{15} - 4 \cdot \frac{15}{15}$

Step 3.12.4.28

Combine $- 4$ and $\frac{15}{15}$ .

$\frac{82}{15} + \frac{- 4 \cdot 15}{15}$

Step 3.12.4.29

Combine the numerators over the common denominator.

$\frac{82 - 4 \cdot 15}{15}$

Step 3.12.4.30

Simplify the numerator.

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Step 3.12.4.30.1

Multiply $- 4$ by $15$ .

$\frac{82 - 60}{15}$

Step 3.12.4.30.2

Subtract $60$ from $82$ .

$\frac{22}{15}$

Step 4

$A r e a = \int_{- 1}^{1} x^{4} - 4 x^{2} + 1 d x - \int_{- 1}^{1} x^{2} - 3 d x$

Step 5

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

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Step 5.1

Combine the integrals into a single integral.

$\int_{- 1}^{1} x^{4} - 4 x^{2} + 1 - (x^{2} - 3) d x$

Step 5.2

Simplify each term.

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Step 5.2.1

Apply the distributive property.

$x^{4} - 4 x^{2} + 1 - x^{2} - - 3$

Step 5.2.2

Multiply $- 1$ by $- 3$ .

$x^{4} - 4 x^{2} + 1 - x^{2} + 3$

$\int_{- 1}^{1} x^{4} - 4 x^{2} + 1 - x^{2} + 3 d x$

Step 5.3

Simplify by adding terms.

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Step 5.3.1

Subtract $x^{2}$ from $- 4 x^{2}$ .

$x^{4} - 5 x^{2} + 1 + 3$

Step 5.3.2

Add $1$ and $3$ .

$x^{4} - 5 x^{2} + 4$

$\int_{- 1}^{1} x^{4} - 5 x^{2} + 4 d x$

Step 5.4

Split the single integral into multiple integrals.

$\int_{- 1}^{1} x^{4} d x + \int_{- 1}^{1} - 5 x^{2} d x + \int_{- 1}^{1} 4 d x$

Step 5.5

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$\frac{1}{5} x^{5}]_{- 1}^{1} + \int_{- 1}^{1} - 5 x^{2} d x + \int_{- 1}^{1} 4 d x$

Step 5.6

Since $- 5$ is constant with respect to $x$ , move $- 5$ out of the integral.

$\frac{1}{5} x^{5}]_{- 1}^{1} - 5 \int_{- 1}^{1} x^{2} d x + \int_{- 1}^{1} 4 d x$

Step 5.7

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

$\frac{1}{5} x^{5}]_{- 1}^{1} - 5 (\frac{1}{3} x^{3}]_{- 1}^{1}) + \int_{- 1}^{1} 4 d x$

Step 5.8

Combine $\frac{1}{3}$ and $x^{3}$ .

$\frac{1}{5} x^{5}]_{- 1}^{1} - 5 (\frac{x^{3}}{3}]_{- 1}^{1}) + \int_{- 1}^{1} 4 d x$

Step 5.9

Apply the constant rule.

$\frac{1}{5} x^{5}]_{- 1}^{1} - 5 (\frac{x^{3}}{3}]_{- 1}^{1}) + 4 x]_{- 1}^{1}$

Step 5.10

Simplify the answer.

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Step 5.10.1

Combine $\frac{1}{5} x^{5}]_{- 1}^{1}$ and $4 x]_{- 1}^{1}$ .

$\frac{1}{5} x^{5} + 4 x]_{- 1}^{1} - 5 (\frac{x^{3}}{3}]_{- 1}^{1})$

Step 5.10.2

Substitute and simplify.

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Step 5.10.2.1

Evaluate $\frac{1}{5} x^{5} + 4 x$ at $1$ and at $- 1$ .

$(\frac{1}{5} \cdot 1^{5} + 4 \cdot 1) - (\frac{1}{5} {(- 1)}^{5} + 4 \cdot - 1) - 5 (\frac{x^{3}}{3}]_{- 1}^{1})$

Step 5.10.2.2

Evaluate $\frac{x^{3}}{3}$ at $1$ and at $- 1$ .

$(\frac{1}{5} \cdot 1^{5} + 4 \cdot 1) - (\frac{1}{5} {(- 1)}^{5} + 4 \cdot - 1) - 5 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3})$

Step 5.10.2.3

Simplify.

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Step 5.10.2.3.1

One to any power is one.

$\frac{1}{5} \cdot 1 + 4 \cdot 1 - (\frac{1}{5} {(- 1)}^{5} + 4 \cdot - 1) - 5 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3})$

Step 5.10.2.3.2

Multiply $\frac{1}{5}$ by $1$ .

$\frac{1}{5} + 4 \cdot 1 - (\frac{1}{5} {(- 1)}^{5} + 4 \cdot - 1) - 5 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3})$

Step 5.10.2.3.3

Multiply $4$ by $1$ .

$\frac{1}{5} + 4 - (\frac{1}{5} {(- 1)}^{5} + 4 \cdot - 1) - 5 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3})$

Step 5.10.2.3.4

To write $4$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{1}{5} + 4 \cdot \frac{5}{5} - (\frac{1}{5} {(- 1)}^{5} + 4 \cdot - 1) - 5 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3})$

Step 5.10.2.3.5

Combine $4$ and $\frac{5}{5}$ .

$\frac{1}{5} + \frac{4 \cdot 5}{5} - (\frac{1}{5} {(- 1)}^{5} + 4 \cdot - 1) - 5 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3})$

Step 5.10.2.3.6

Combine the numerators over the common denominator.

$\frac{1 + 4 \cdot 5}{5} - (\frac{1}{5} {(- 1)}^{5} + 4 \cdot - 1) - 5 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3})$

Step 5.10.2.3.7

Simplify the numerator.

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Step 5.10.2.3.7.1

Multiply $4$ by $5$ .

$\frac{1 + 20}{5} - (\frac{1}{5} {(- 1)}^{5} + 4 \cdot - 1) - 5 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3})$

Step 5.10.2.3.7.2

Add $1$ and $20$ .

$\frac{21}{5} - (\frac{1}{5} {(- 1)}^{5} + 4 \cdot - 1) - 5 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3})$

Step 5.10.2.3.8

Raise $- 1$ to the power of $5$ .

$\frac{21}{5} - (\frac{1}{5} \cdot - 1 + 4 \cdot - 1) - 5 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3})$

Step 5.10.2.3.9

Combine $\frac{1}{5}$ and $- 1$ .

$\frac{21}{5} - (\frac{- 1}{5} + 4 \cdot - 1) - 5 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3})$

Step 5.10.2.3.10

Move the negative in front of the fraction.

$\frac{21}{5} - (- \frac{1}{5} + 4 \cdot - 1) - 5 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3})$

Step 5.10.2.3.11

Multiply $4$ by $- 1$ .

$\frac{21}{5} - (- \frac{1}{5} - 4) - 5 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3})$

Step 5.10.2.3.12

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{21}{5} - (- \frac{1}{5} - 4 \cdot \frac{5}{5}) - 5 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3})$

Step 5.10.2.3.13

Combine $- 4$ and $\frac{5}{5}$ .

$\frac{21}{5} - (- \frac{1}{5} + \frac{- 4 \cdot 5}{5}) - 5 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3})$

Step 5.10.2.3.14

Combine the numerators over the common denominator.

$\frac{21}{5} - \frac{- 1 - 4 \cdot 5}{5} - 5 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3})$

Step 5.10.2.3.15

Simplify the numerator.

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Step 5.10.2.3.15.1

Multiply $- 4$ by $5$ .

$\frac{21}{5} - \frac{- 1 - 20}{5} - 5 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3})$

Step 5.10.2.3.15.2

Subtract $20$ from $- 1$ .

$\frac{21}{5} - \frac{- 21}{5} - 5 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3})$

Step 5.10.2.3.16

Move the negative in front of the fraction.

$\frac{21}{5} - - \frac{21}{5} - 5 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3})$

Step 5.10.2.3.17

Multiply $- 1$ by $- 1$ .

$\frac{21}{5} + 1 (\frac{21}{5}) - 5 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3})$

Step 5.10.2.3.18

Multiply $\frac{21}{5}$ by $1$ .

$\frac{21}{5} + \frac{21}{5} - 5 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3})$

Step 5.10.2.3.19

Combine the numerators over the common denominator.

$\frac{21 + 21}{5} - 5 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3})$

Step 5.10.2.3.20

Add $21$ and $21$ .

$\frac{42}{5} - 5 ((\frac{1^{3}}{3}) - \frac{{(- 1)}^{3}}{3})$

Step 5.10.2.3.21

One to any power is one.

$\frac{42}{5} - 5 (\frac{1}{3} - \frac{{(- 1)}^{3}}{3})$

Step 5.10.2.3.22

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

$\frac{42}{5} - 5 (\frac{1}{3} - \frac{- 1}{3})$

Step 5.10.2.3.23

Move the negative in front of the fraction.

$\frac{42}{5} - 5 (\frac{1}{3} - - \frac{1}{3})$

Step 5.10.2.3.24

Multiply $- 1$ by $- 1$ .

$\frac{42}{5} - 5 (\frac{1}{3} + 1 (\frac{1}{3}))$

Step 5.10.2.3.25

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

$\frac{42}{5} - 5 (\frac{1}{3} + \frac{1}{3})$

Step 5.10.2.3.26

Combine the numerators over the common denominator.

$\frac{42}{5} - 5 \frac{1 + 1}{3}$

Step 5.10.2.3.27

Add $1$ and $1$ .

$\frac{42}{5} - 5 (\frac{2}{3})$

Step 5.10.2.3.28

Combine $- 5$ and $\frac{2}{3}$ .

$\frac{42}{5} + \frac{- 5 \cdot 2}{3}$

Step 5.10.2.3.29

Multiply $- 5$ by $2$ .

$\frac{42}{5} + \frac{- 10}{3}$

Step 5.10.2.3.30

Move the negative in front of the fraction.

$\frac{42}{5} - \frac{10}{3}$

Step 5.10.2.3.31

To write $\frac{42}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{42}{5} \cdot \frac{3}{3} - \frac{10}{3}$

Step 5.10.2.3.32

To write $- \frac{10}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{42}{5} \cdot \frac{3}{3} - \frac{10}{3} \cdot \frac{5}{5}$

Step 5.10.2.3.33

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.10.2.3.33.1

Multiply $\frac{42}{5}$ by $\frac{3}{3}$ .

$\frac{42 \cdot 3}{5 \cdot 3} - \frac{10}{3} \cdot \frac{5}{5}$

Step 5.10.2.3.33.2

Multiply $5$ by $3$ .

$\frac{42 \cdot 3}{15} - \frac{10}{3} \cdot \frac{5}{5}$

Step 5.10.2.3.33.3

Multiply $\frac{10}{3}$ by $\frac{5}{5}$ .

$\frac{42 \cdot 3}{15} - \frac{10 \cdot 5}{3 \cdot 5}$

Step 5.10.2.3.33.4

Multiply $3$ by $5$ .

$\frac{42 \cdot 3}{15} - \frac{10 \cdot 5}{15}$

Step 5.10.2.3.34

Combine the numerators over the common denominator.

$\frac{42 \cdot 3 - 10 \cdot 5}{15}$

Step 5.10.2.3.35

Simplify the numerator.

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Step 5.10.2.3.35.1

Multiply $42$ by $3$ .

$\frac{126 - 10 \cdot 5}{15}$

Step 5.10.2.3.35.2

Multiply $- 10$ by $5$ .

$\frac{126 - 50}{15}$

Step 5.10.2.3.35.3

Subtract $50$ from $126$ .

$\frac{76}{15}$

Step 6

$A r e a = \int_{1}^{2} x^{2} - 3 d x - \int_{1}^{2} x^{4} - 4 x^{2} + 1 d x$

Step 7

Integrate to find the area between $1$ and $2$ .

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Step 7.1

Combine the integrals into a single integral.

$\int_{1}^{2} x^{2} - 3 - (x^{4} - 4 x^{2} + 1) d x$

Step 7.2

Simplify each term.

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Step 7.2.1

Apply the distributive property.

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

Step 7.2.2

Simplify.

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Step 7.2.2.1

Multiply $- 4$ by $- 1$ .

$x^{2} - 3 - x^{4} + 4 x^{2} - 1 \cdot 1$

Step 7.2.2.2

Multiply $- 1$ by $1$ .

$x^{2} - 3 - x^{4} + 4 x^{2} - 1$

$\int_{1}^{2} x^{2} - 3 - x^{4} + 4 x^{2} - 1 d x$

Step 7.3

Simplify by adding terms.

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Step 7.3.1

Add $x^{2}$ and $4 x^{2}$ .

$5 x^{2} - 3 - x^{4} - 1$

Step 7.3.2

Subtract $1$ from $- 3$ .

$5 x^{2} - x^{4} - 4$

$\int_{1}^{2} 5 x^{2} - x^{4} - 4 d x$

Step 7.4

Split the single integral into multiple integrals.

$\int_{1}^{2} 5 x^{2} d x + \int_{1}^{2} - x^{4} d x + \int_{1}^{2} - 4 d x$

Step 7.5

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int_{1}^{2} x^{2} d x + \int_{1}^{2} - x^{4} d x + \int_{1}^{2} - 4 d x$

Step 7.6

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

$5 (\frac{1}{3} x^{3}]_{1}^{2}) + \int_{1}^{2} - x^{4} d x + \int_{1}^{2} - 4 d x$

Step 7.7

Combine $\frac{1}{3}$ and $x^{3}$ .

$5 (\frac{x^{3}}{3}]_{1}^{2}) + \int_{1}^{2} - x^{4} d x + \int_{1}^{2} - 4 d x$

Step 7.8

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

$5 (\frac{x^{3}}{3}]_{1}^{2}) - \int_{1}^{2} x^{4} d x + \int_{1}^{2} - 4 d x$

Step 7.9

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$5 (\frac{x^{3}}{3}]_{1}^{2}) - (\frac{1}{5} x^{5}]_{1}^{2}) + \int_{1}^{2} - 4 d x$

Step 7.10

Combine $\frac{1}{5}$ and $x^{5}$ .

$5 (\frac{x^{3}}{3}]_{1}^{2}) - (\frac{x^{5}}{5}]_{1}^{2}) + \int_{1}^{2} - 4 d x$

Step 7.11

Apply the constant rule.

$5 (\frac{x^{3}}{3}]_{1}^{2}) - (\frac{x^{5}}{5}]_{1}^{2}) + - 4 x]_{1}^{2}$

Step 7.12

Substitute and simplify.

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Step 7.12.1

Evaluate $\frac{x^{3}}{3}$ at $2$ and at $1$ .

$5 ((\frac{2^{3}}{3}) - \frac{1^{3}}{3}) - (\frac{x^{5}}{5}]_{1}^{2}) + - 4 x]_{1}^{2}$

Step 7.12.2

Evaluate $\frac{x^{5}}{5}$ at $2$ and at $1$ .

$5 (\frac{2^{3}}{3} - \frac{1^{3}}{3}) - (\frac{2^{5}}{5} - \frac{1^{5}}{5}) + - 4 x]_{1}^{2}$

Step 7.12.3

Evaluate $- 4 x$ at $2$ and at $1$ .

$5 (\frac{2^{3}}{3} - \frac{1^{3}}{3}) - (\frac{2^{5}}{5} - \frac{1^{5}}{5}) + (- 4 \cdot 2) + 4 \cdot 1$

Step 7.12.4

Simplify.

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Step 7.12.4.1

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

$5 (\frac{8}{3} - \frac{1^{3}}{3}) - (\frac{2^{5}}{5} - \frac{1^{5}}{5}) + (- 4 \cdot 2) + 4 \cdot 1$

Step 7.12.4.2

One to any power is one.

$5 (\frac{8}{3} - \frac{1}{3}) - (\frac{2^{5}}{5} - \frac{1^{5}}{5}) + (- 4 \cdot 2) + 4 \cdot 1$

Step 7.12.4.3

Combine the numerators over the common denominator.

$5 \frac{8 - 1}{3} - (\frac{2^{5}}{5} - \frac{1^{5}}{5}) + (- 4 \cdot 2) + 4 \cdot 1$

Step 7.12.4.4

Subtract $1$ from $8$ .

$5 (\frac{7}{3}) - (\frac{2^{5}}{5} - \frac{1^{5}}{5}) + (- 4 \cdot 2) + 4 \cdot 1$

Step 7.12.4.5

Combine $5$ and $\frac{7}{3}$ .

$\frac{5 \cdot 7}{3} - (\frac{2^{5}}{5} - \frac{1^{5}}{5}) + (- 4 \cdot 2) + 4 \cdot 1$

Step 7.12.4.6

Multiply $5$ by $7$ .

$\frac{35}{3} - (\frac{2^{5}}{5} - \frac{1^{5}}{5}) + (- 4 \cdot 2) + 4 \cdot 1$

Step 7.12.4.7

Raise $2$ to the power of $5$ .

$\frac{35}{3} - (\frac{32}{5} - \frac{1^{5}}{5}) + (- 4 \cdot 2) + 4 \cdot 1$

Step 7.12.4.8

One to any power is one.

$\frac{35}{3} - (\frac{32}{5} - \frac{1}{5}) + (- 4 \cdot 2) + 4 \cdot 1$

Step 7.12.4.9

Combine the numerators over the common denominator.

$\frac{35}{3} - \frac{32 - 1}{5} + (- 4 \cdot 2) + 4 \cdot 1$

Step 7.12.4.10

Subtract $1$ from $32$ .

$\frac{35}{3} - \frac{31}{5} + (- 4 \cdot 2) + 4 \cdot 1$

Step 7.12.4.11

To write $\frac{35}{3}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{35}{3} \cdot \frac{5}{5} - \frac{31}{5} + (- 4 \cdot 2) + 4 \cdot 1$

Step 7.12.4.12

To write $- \frac{31}{5}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{35}{3} \cdot \frac{5}{5} - \frac{31}{5} \cdot \frac{3}{3} + (- 4 \cdot 2) + 4 \cdot 1$

Step 7.12.4.13

Write each expression with a common denominator of $15$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.12.4.13.1

Multiply $\frac{35}{3}$ by $\frac{5}{5}$ .

$\frac{35 \cdot 5}{3 \cdot 5} - \frac{31}{5} \cdot \frac{3}{3} + (- 4 \cdot 2) + 4 \cdot 1$

Step 7.12.4.13.2

Multiply $3$ by $5$ .

$\frac{35 \cdot 5}{15} - \frac{31}{5} \cdot \frac{3}{3} + (- 4 \cdot 2) + 4 \cdot 1$

Step 7.12.4.13.3

Multiply $\frac{31}{5}$ by $\frac{3}{3}$ .

$\frac{35 \cdot 5}{15} - \frac{31 \cdot 3}{5 \cdot 3} + (- 4 \cdot 2) + 4 \cdot 1$

Step 7.12.4.13.4

Multiply $5$ by $3$ .

$\frac{35 \cdot 5}{15} - \frac{31 \cdot 3}{15} + (- 4 \cdot 2) + 4 \cdot 1$

Step 7.12.4.14

Combine the numerators over the common denominator.

$\frac{35 \cdot 5 - 31 \cdot 3}{15} + (- 4 \cdot 2) + 4 \cdot 1$

Step 7.12.4.15

Simplify the numerator.

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Step 7.12.4.15.1

Multiply $35$ by $5$ .

$\frac{175 - 31 \cdot 3}{15} + (- 4 \cdot 2) + 4 \cdot 1$

Step 7.12.4.15.2

Multiply $- 31$ by $3$ .

$\frac{175 - 93}{15} + (- 4 \cdot 2) + 4 \cdot 1$

Step 7.12.4.15.3

Subtract $93$ from $175$ .

$\frac{82}{15} + (- 4 \cdot 2) + 4 \cdot 1$

Step 7.12.4.16

Multiply $- 4$ by $2$ .

$\frac{82}{15} - 8 + 4 \cdot 1$

Step 7.12.4.17

Multiply $4$ by $1$ .

$\frac{82}{15} - 8 + 4$

Step 7.12.4.18

Add $- 8$ and $4$ .

$\frac{82}{15} - 4$

Step 7.12.4.19

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

$\frac{82}{15} - 4 \cdot \frac{15}{15}$

Step 7.12.4.20

Combine $- 4$ and $\frac{15}{15}$ .

$\frac{82}{15} + \frac{- 4 \cdot 15}{15}$

Step 7.12.4.21

Combine the numerators over the common denominator.

$\frac{82 - 4 \cdot 15}{15}$

Step 7.12.4.22

Simplify the numerator.

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Step 7.12.4.22.1

Multiply $- 4$ by $15$ .

$\frac{82 - 60}{15}$

Step 7.12.4.22.2

Subtract $60$ from $82$ .

$\frac{22}{15}$

Step 8

Add the areas $\frac{22}{15} + \frac{76}{15} + \frac{22}{15}$ .

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Step 8.1

Combine the numerators over the common denominator.

$\frac{22 + 76 + 22}{15}$

Step 8.2

Simplify the expression.

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Step 8.2.1

Add $22$ and $76$ .

$\frac{98 + 22}{15}$

Step 8.2.2

Add $98$ and $22$ .

$\frac{120}{15}$

Step 8.2.3

Divide $120$ by $15$ .

$8$

Step 9