Calculus Examples

$y = \frac{x^{4}}{4} + \frac{1}{8 x^{2}}$ ; $[1, 4]$

Step 1

Solve by substitution to find the intersection between the curves.

Tap for more steps...

Step 1.1

Eliminate the equal sides of each equation and combine.

$\frac{x^{4}}{4} + \frac{1}{8 x^{2}} = 0$

Step 1.2

Solve $\frac{x^{4}}{4} + \frac{1}{8 x^{2}} = 0$ for $x$ .

Tap for more steps...

Step 1.2.1

Find the LCD of the terms in the equation.

Tap for more steps...

Step 1.2.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$4, 8 x^{2}, 1 + y = 0$

Step 1.2.1.2

Since $4, 8 x^{2}, 1$ contains both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $4, 8, 1$ then find LCM for the variable part $x^{2}$ .

Step 1.2.1.3

The LCM is the smallest positive number that all of the numbers divide into evenly.

$1$ List the prime factors of each number.

Step 1.2.1.4

$4$ has factors of $2$ and $2$ .

$2 \cdot 2 + y = 0$

Step 1.2.1.5

The prime factors for $8$ are $2 \cdot 2 \cdot 2$ .

Tap for more steps...

Step 1.2.1.5.1

$8$ has factors of $2$ and $4$ .

$2 \cdot 4$

Step 1.2.1.5.2

$4$ has factors of $2$ and $2$ .

$2 \cdot 2 \cdot 2$

$2 \cdot 2 \cdot 2 + y = 0$

Step 1.2.1.6

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not $p r i m e + y = 0$

Step 1.2.1.7

The LCM of $4, 8, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$2 \cdot 2 \cdot 2 + y = 0$

Step 1.2.1.8

Multiply $2 \cdot 2 \cdot 2$ .

Tap for more steps...

Step 1.2.1.8.1

Multiply $2$ by $2$ .

$4 \cdot 2 + y = 0$

Step 1.2.1.8.2

Multiply $4$ by $2$ .

$8 + y = 0$

Step 1.2.1.9

The factors for $x^{2}$ are $x \cdot x$ , which is $x$ multiplied by each other $2$ times.

$x^{2} = x \cdot x$

Step 1.2.1.10

The LCM of $x^{2}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x \cdot x + y = 0$

Step 1.2.1.11

Multiply $x$ by $x$ .

$x^{2} + y = 0$

Step 1.2.1.12

The LCM for $4, 8 x^{2}, 1$ is the numeric part $8$ multiplied by the variable part.

$8 x^{2} + y = 0$

Step 1.2.2

Multiply each term in $\frac{x^{4}}{4} + \frac{1}{8 x^{2}} = 0$ by $8 x^{2}$ to eliminate the fractions.

Tap for more steps...

Step 1.2.2.1

Multiply each term in $\frac{x^{4}}{4} + \frac{1}{8 x^{2}} = 0$ by $8 x^{2}$ .

$\frac{x^{4}}{4} (8 x^{2}) + \frac{1}{8 x^{2}} (8 x^{2}) = 0 (8 x^{2})$

Step 1.2.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.2.2.1

Simplify each term.

Tap for more steps...

Step 1.2.2.2.1.1

Rewrite using the commutative property of multiplication.

$8 \frac{x^{4}}{4} x^{2} + \frac{1}{8 x^{2}} (8 x^{2}) = 0 (8 x^{2})$

Step 1.2.2.2.1.2

Cancel the common factor of $4$ .

Tap for more steps...

Step 1.2.2.2.1.2.1

Factor $4$ out of $8$ .

$4 (2) \frac{x^{4}}{4} x^{2} + \frac{1}{8 x^{2}} (8 x^{2}) = 0 (8 x^{2})$

Step 1.2.2.2.1.2.2

Cancel the common factor.

$4 \cdot 2 \frac{x^{4}}{4} x^{2} + \frac{1}{8 x^{2}} (8 x^{2}) = 0 (8 x^{2})$

Step 1.2.2.2.1.2.3

Rewrite the expression.

$2 x^{4} x^{2} + \frac{1}{8 x^{2}} (8 x^{2}) = 0 (8 x^{2})$

Step 1.2.2.2.1.3

Multiply $x^{4}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 1.2.2.2.1.3.1

Move $x^{2}$ .

$2 (x^{2} x^{4}) + \frac{1}{8 x^{2}} (8 x^{2}) = 0 (8 x^{2})$

Step 1.2.2.2.1.3.2

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

$2 x^{2 + 4} + \frac{1}{8 x^{2}} (8 x^{2}) = 0 (8 x^{2})$

Step 1.2.2.2.1.3.3

Add $2$ and $4$ .

$2 x^{6} + \frac{1}{8 x^{2}} (8 x^{2}) = 0 (8 x^{2})$

Step 1.2.2.2.1.4

Rewrite using the commutative property of multiplication.

$2 x^{6} + 8 \frac{1}{8 x^{2}} x^{2} = 0 (8 x^{2})$

Step 1.2.2.2.1.5

Cancel the common factor of $8$ .

Tap for more steps...

Step 1.2.2.2.1.5.1

Factor $8$ out of $8 x^{2}$ .

$2 x^{6} + 8 \frac{1}{8 (x^{2})} x^{2} = 0 (8 x^{2})$

Step 1.2.2.2.1.5.2

Cancel the common factor.

$2 x^{6} + 8 \frac{1}{8 x^{2}} x^{2} = 0 (8 x^{2})$

Step 1.2.2.2.1.5.3

Rewrite the expression.

$2 x^{6} + \frac{1}{x^{2}} x^{2} = 0 (8 x^{2})$

Step 1.2.2.2.1.6

Cancel the common factor of $x^{2}$ .

Tap for more steps...

Step 1.2.2.2.1.6.1

Cancel the common factor.

$2 x^{6} + \frac{1}{x^{2}} x^{2} = 0 (8 x^{2})$

Step 1.2.2.2.1.6.2

Rewrite the expression.

$2 x^{6} + 1 = 0 (8 x^{2})$

Step 1.2.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.2.3.1

Multiply $0 (8 x^{2})$ .

Tap for more steps...

Step 1.2.2.3.1.1

Multiply $8$ by $0$ .

$2 x^{6} + 1 = 0 x^{2}$

Step 1.2.2.3.1.2

Multiply $0$ by $x^{2}$ .

$2 x^{6} + 1 = 0$

Step 1.2.3

Solve the equation.

Tap for more steps...

Step 1.2.3.1

Subtract $1$ from both sides of the equation.

$2 x^{6} = - 1$

Step 1.2.3.2

Divide each term in $2 x^{6} = - 1$ by $2$ and simplify.

Tap for more steps...

Step 1.2.3.2.1

Divide each term in $2 x^{6} = - 1$ by $2$ .

$\frac{2 x^{6}}{2} = \frac{- 1}{2}$

Step 1.2.3.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.3.2.2.1

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.3.2.2.1.1

Cancel the common factor.

$\frac{2 x^{6}}{2} = \frac{- 1}{2}$

Step 1.2.3.2.2.1.2

Divide $x^{6}$ by $1$ .

$x^{6} = \frac{- 1}{2}$

Step 1.2.3.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.3.2.3.1

Move the negative in front of the fraction.

$x^{6} = - \frac{1}{2}$

Step 1.2.3.3

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

$x = \pm \sqrt[6]{- \frac{1}{2}}$

Step 1.2.3.4

Simplify $\pm \sqrt[6]{- \frac{1}{2}}$ .

Tap for more steps...

Step 1.2.3.4.1

Rewrite $- \frac{1}{2}$ as $i^{6} \frac{1^{6}}{2}$ .

Tap for more steps...

Step 1.2.3.4.1.1

Rewrite $- 1$ as $i^{6}$ .

$x = \pm \sqrt[6]{i^{6} \frac{1}{2}}$

Step 1.2.3.4.1.2

Rewrite $1$ as $1^{6}$ .

$x = \pm \sqrt[6]{i^{6} \frac{1^{6}}{2}}$

Step 1.2.3.4.2

Pull terms out from under the radical.

$x = \pm i \sqrt[6]{\frac{1^{6}}{2}}$

Step 1.2.3.4.3

One to any power is one.

$x = \pm i \sqrt[6]{\frac{1}{2}}$

Step 1.2.3.4.4

Rewrite $\sqrt[6]{\frac{1}{2}}$ as $\frac{\sqrt[6]{1}}{\sqrt[6]{2}}$ .

$x = \pm i \frac{\sqrt[6]{1}}{\sqrt[6]{2}}$

Step 1.2.3.4.5

Any root of $1$ is $1$ .

$x = \pm i \frac{1}{\sqrt[6]{2}}$

Step 1.2.3.4.6

Combine $i$ and $\frac{1}{\sqrt[6]{2}}$ .

$x = \pm \frac{i}{\sqrt[6]{2}}$

Step 1.2.3.5

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

Tap for more steps...

Step 1.2.3.5.1

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

$x = \frac{i}{\sqrt[6]{2}}$

Step 1.2.3.5.2

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

$x = - \frac{i}{\sqrt[6]{2}}$

Step 1.2.3.5.3

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

$x = \frac{i}{\sqrt[6]{2}}, - \frac{i}{\sqrt[6]{2}}$

Step 1.3

Substitute $\frac{i}{\sqrt[6]{2}}$ for $x$ .

$y = 0$

Step 1.4

List all of the solutions.

$y = 0, x = \frac{i}{\sqrt[6]{2}}$

$y = 0, x = - \frac{i}{\sqrt[6]{2}}$

$y = 0, x = \frac{i}{\sqrt[6]{2}}$

$y = 0, x = - \frac{i}{\sqrt[6]{2}}$

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.

$A r e a = \int_{1}^{4} \frac{x^{4}}{4} + \frac{1}{8 x^{2}} d x - \int_{1}^{4} 0 d x$

Step 3

Integrate to find the area between $1$ and $4$ .

Tap for more steps...

Step 3.1

Combine the integrals into a single integral.

$\int_{1}^{4} \frac{x^{4}}{4} + \frac{1}{8 x^{2}} - (0) d x$

Step 3.2

Subtract $0$ from $\frac{x^{4}}{4} + \frac{1}{8 x^{2}}$ .

$\int_{1}^{4} \frac{x^{4}}{4} + \frac{1}{8 x^{2}} d x$

Step 3.3

Split the single integral into multiple integrals.

$\int_{1}^{4} \frac{x^{4}}{4} d x + \int_{1}^{4} \frac{1}{8 x^{2}} d x$

Step 3.4

Since $\frac{1}{4}$ is constant with respect to $x$ , move $\frac{1}{4}$ out of the integral.

$\frac{1}{4} \int_{1}^{4} x^{4} d x + \int_{1}^{4} \frac{1}{8 x^{2}} d x$

Step 3.5

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

$\frac{1}{4} \frac{1}{5} x^{5}]_{1}^{4} + \int_{1}^{4} \frac{1}{8 x^{2}} d x$

Step 3.6

Since $\frac{1}{8}$ is constant with respect to $x$ , move $\frac{1}{8}$ out of the integral.

$\frac{1}{4} \frac{1}{5} x^{5}]_{1}^{4} + \frac{1}{8} \int_{1}^{4} \frac{1}{x^{2}} d x$

Step 3.7

Apply basic rules of exponents.

Tap for more steps...

Step 3.7.1

Move $x^{2}$ out of the denominator by raising it to the $- 1$ power.

$\frac{1}{4} \frac{1}{5} x^{5}]_{1}^{4} + \frac{1}{8} \int_{1}^{4} {(x^{2})}^{- 1} d x$

Step 3.7.2

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

Tap for more steps...

Step 3.7.2.1

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

$\frac{1}{4} \frac{1}{5} x^{5}]_{1}^{4} + \frac{1}{8} \int_{1}^{4} x^{2 \cdot - 1} d x$

Step 3.7.2.2

Multiply $2$ by $- 1$ .

$\frac{1}{4} \frac{1}{5} x^{5}]_{1}^{4} + \frac{1}{8} \int_{1}^{4} x^{- 2} d x$

Step 3.8

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

$\frac{1}{4} \frac{1}{5} x^{5}]_{1}^{4} + \frac{1}{8} - x^{- 1}]_{1}^{4}$

Step 3.9

Substitute and simplify.

Tap for more steps...

Step 3.9.1

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

$\frac{1}{4} ((\frac{1}{5} \cdot 4^{5}) - \frac{1}{5} \cdot 1^{5}) + \frac{1}{8} - x^{- 1}]_{1}^{4}$

Step 3.9.2

Evaluate $- x^{- 1}$ at $4$ and at $1$ .

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

Step 3.9.3

Simplify.

Tap for more steps...

Step 3.9.3.1

Raise $4$ to the power of $5$ .

$\frac{1}{4} (\frac{1}{5} \cdot 1024 - \frac{1}{5} \cdot 1^{5}) + \frac{1}{8} (- 4^{- 1} + 1^{- 1})$

Step 3.9.3.2

Combine $\frac{1}{5}$ and $1024$ .

$\frac{1}{4} (\frac{1024}{5} - \frac{1}{5} \cdot 1^{5}) + \frac{1}{8} (- 4^{- 1} + 1^{- 1})$

Step 3.9.3.3

One to any power is one.

$\frac{1}{4} (\frac{1024}{5} - \frac{1}{5} \cdot 1) + \frac{1}{8} (- 4^{- 1} + 1^{- 1})$

Step 3.9.3.4

Multiply $- 1$ by $1$ .

$\frac{1}{4} (\frac{1024}{5} - \frac{1}{5}) + \frac{1}{8} (- 4^{- 1} + 1^{- 1})$

Step 3.9.3.5

Combine the numerators over the common denominator.

$\frac{1}{4} \cdot \frac{1024 - 1}{5} + \frac{1}{8} (- 4^{- 1} + 1^{- 1})$

Step 3.9.3.6

Subtract $1$ from $1024$ .

$\frac{1}{4} \cdot \frac{1023}{5} + \frac{1}{8} (- 4^{- 1} + 1^{- 1})$

Step 3.9.3.7

Multiply $\frac{1}{4}$ by $\frac{1023}{5}$ .

$\frac{1023}{4 \cdot 5} + \frac{1}{8} (- 4^{- 1} + 1^{- 1})$

Step 3.9.3.8

Multiply $4$ by $5$ .

$\frac{1023}{20} + \frac{1}{8} (- 4^{- 1} + 1^{- 1})$

Step 3.9.3.9

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1023}{20} + \frac{1}{8} (- \frac{1}{4} + 1^{- 1})$

Step 3.9.3.10

One to any power is one.

$\frac{1023}{20} + \frac{1}{8} (- \frac{1}{4} + 1)$

Step 3.9.3.11

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

$\frac{1023}{20} + \frac{1}{8} (- \frac{1}{4} + \frac{4}{4})$

Step 3.9.3.12

Combine the numerators over the common denominator.

$\frac{1023}{20} + \frac{1}{8} \cdot \frac{- 1 + 4}{4}$

Step 3.9.3.13

Add $- 1$ and $4$ .

$\frac{1023}{20} + \frac{1}{8} \cdot \frac{3}{4}$

Step 3.9.3.14

Multiply $\frac{1}{8}$ by $\frac{3}{4}$ .

$\frac{1023}{20} + \frac{3}{8 \cdot 4}$

Step 3.9.3.15

Multiply $8$ by $4$ .

$\frac{1023}{20} + \frac{3}{32}$

Step 3.9.3.16

To write $\frac{1023}{20}$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$\frac{1023}{20} \cdot \frac{8}{8} + \frac{3}{32}$

Step 3.9.3.17

To write $\frac{3}{32}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{1023}{20} \cdot \frac{8}{8} + \frac{3}{32} \cdot \frac{5}{5}$

Step 3.9.3.18

Write each expression with a common denominator of $160$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 3.9.3.18.1

Multiply $\frac{1023}{20}$ by $\frac{8}{8}$ .

$\frac{1023 \cdot 8}{20 \cdot 8} + \frac{3}{32} \cdot \frac{5}{5}$

Step 3.9.3.18.2

Multiply $20$ by $8$ .

$\frac{1023 \cdot 8}{160} + \frac{3}{32} \cdot \frac{5}{5}$

Step 3.9.3.18.3

Multiply $\frac{3}{32}$ by $\frac{5}{5}$ .

$\frac{1023 \cdot 8}{160} + \frac{3 \cdot 5}{32 \cdot 5}$

Step 3.9.3.18.4

Multiply $32$ by $5$ .

$\frac{1023 \cdot 8}{160} + \frac{3 \cdot 5}{160}$

Step 3.9.3.19

Combine the numerators over the common denominator.

$\frac{1023 \cdot 8 + 3 \cdot 5}{160}$

Step 3.9.3.20

Simplify the numerator.

Tap for more steps...

Step 3.9.3.20.1

Multiply $1023$ by $8$ .

$\frac{8184 + 3 \cdot 5}{160}$

Step 3.9.3.20.2

Multiply $3$ by $5$ .

$\frac{8184 + 15}{160}$

Step 3.9.3.20.3

Add $8184$ and $15$ .

$\frac{8199}{160}$

Step 4