Calculus Examples

$y = x$ , $y = x^{3}$

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.

$x = x^{3}$

Step 1.2

Solve $x = x^{3}$ for $x$ .

Tap for more steps...

Step 1.2.1

Subtract $x^{3}$ from both sides of the equation.

$x - x^{3} = 0$

Step 1.2.2

Factor the left side of the equation.

Tap for more steps...

Step 1.2.2.1

Factor $x$ out of $x - x^{3}$ .

Tap for more steps...

Step 1.2.2.1.1

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

$x - x^{3} = 0$

Step 1.2.2.1.2

Factor $x$ out of $x^{1}$ .

$x \cdot 1 - x^{3} = 0$

Step 1.2.2.1.3

Factor $x$ out of $- x^{3}$ .

$x \cdot 1 + x (- x^{2}) = 0$

Step 1.2.2.1.4

Factor $x$ out of $x \cdot 1 + x (- x^{2})$ .

$x (1 - x^{2}) = 0$

Step 1.2.2.2

Rewrite $1$ as $1^{2}$ .

$x (1^{2} - x^{2}) = 0$

Step 1.2.2.3

Factor.

Tap for more steps...

Step 1.2.2.3.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = x$ .

$x ((1 + x) (1 - x)) = 0$

Step 1.2.2.3.2

Remove unnecessary parentheses.

$x (1 + x) (1 - x) = 0$

Step 1.2.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$1 + x = 0$

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

Step 1.2.4

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.5

Set $1 + x$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 1.2.5.1

Set $1 + x$ equal to $0$ .

$1 + x = 0$

Step 1.2.5.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 1.2.6

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

Tap for more steps...

Step 1.2.6.1

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

$1 - x = 0$

Step 1.2.6.2

Solve $1 - x = 0$ for $x$ .

Tap for more steps...

Step 1.2.6.2.1

Subtract $1$ from both sides of the equation.

$- x = - 1$

Step 1.2.6.2.2

Divide each term in $- x = - 1$ by $- 1$ and simplify.

Tap for more steps...

Step 1.2.6.2.2.1

Divide each term in $- x = - 1$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 1}{- 1}$

Step 1.2.6.2.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.6.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 1}{- 1}$

Step 1.2.6.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{- 1}{- 1}$

Step 1.2.6.2.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.6.2.2.3.1

Divide $- 1$ by $- 1$ .

$x = 1$

Step 1.2.7

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

$x = 0, - 1, 1$

Step 1.3

Evaluate $y$ when $x = 0$ .

Tap for more steps...

Step 1.3.1

Substitute $0$ for $x$ .

$y = {(0)}^{3}$

Step 1.3.2

Substitute $0$ for $x$ in $y = {(0)}^{3}$ and solve for $y$ .

Tap for more steps...

Step 1.3.2.1

Remove parentheses.

$y = 0^{3}$

Step 1.3.2.2

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

$y = 0$

Step 1.4

Evaluate $y$ when $x = - 1$ .

Tap for more steps...

Step 1.4.1

Substitute $- 1$ for $x$ .

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

Step 1.4.2

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

$y = - 1$

Step 1.5

Evaluate $y$ when $x = 1$ .

Tap for more steps...

Step 1.5.1

Substitute $1$ for $x$ .

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

Step 1.5.2

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

Tap for more steps...

Step 1.5.2.1

Remove parentheses.

$y = 1^{3}$

Step 1.5.2.2

One to any power is one.

$y = 1$

Step 1.6

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

$(0, 0)$

$(- 1, - 1)$

$(1, 1)$

$(0, 0)$

$(- 1, - 1)$

$(1, 1)$

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.

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

Step 3

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

Tap for more steps...

Step 3.1

Combine the integrals into a single integral.

$\int_{- 1}^{0} x^{3} - (x) d x$

Step 3.2

Multiply $- 1$ by $x$ .

$\int_{- 1}^{0} x^{3} - x d x$

Step 3.3

Split the single integral into multiple integrals.

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

Step 3.4

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

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

Step 3.5

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

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

Step 3.6

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

$\frac{1}{4} x^{4}]_{- 1}^{0} - (\frac{1}{2} x^{2}]_{- 1}^{0})$

Step 3.7

Simplify the answer.

Tap for more steps...

Step 3.7.1

Combine $\frac{1}{2}$ and $x^{2}$ .

$\frac{1}{4} x^{4}]_{- 1}^{0} - (\frac{x^{2}}{2}]_{- 1}^{0})$

Step 3.7.2

Substitute and simplify.

Tap for more steps...

Step 3.7.2.1

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

$(\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 1)}^{4} - (\frac{x^{2}}{2}]_{- 1}^{0})$

Step 3.7.2.2

Evaluate $\frac{x^{2}}{2}$ at $0$ and at $- 1$ .

$\frac{1}{4} \cdot 0^{4} - \frac{1}{4} {(- 1)}^{4} - (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 3.7.2.3

Simplify.

Tap for more steps...

Step 3.7.2.3.1

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

$\frac{1}{4} \cdot 0 - \frac{1}{4} {(- 1)}^{4} - (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 3.7.2.3.2

Multiply $\frac{1}{4}$ by $0$ .

$0 - \frac{1}{4} {(- 1)}^{4} - (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 3.7.2.3.3

Raise $- 1$ to the power of $4$ .

$0 - \frac{1}{4} \cdot 1 - (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 3.7.2.3.4

Multiply $- 1$ by $1$ .

$0 - \frac{1}{4} - (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 3.7.2.3.5

Subtract $\frac{1}{4}$ from $0$ .

$- \frac{1}{4} - (\frac{0^{2}}{2} - \frac{{(- 1)}^{2}}{2})$

Step 3.7.2.3.6

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

$- \frac{1}{4} - (\frac{0}{2} - \frac{{(- 1)}^{2}}{2})$

Step 3.7.2.3.7

Cancel the common factor of $0$ and $2$ .

Tap for more steps...

Step 3.7.2.3.7.1

Factor $2$ out of $0$ .

$- \frac{1}{4} - (\frac{2 (0)}{2} - \frac{{(- 1)}^{2}}{2})$

Step 3.7.2.3.7.2

Cancel the common factors.

Tap for more steps...

Step 3.7.2.3.7.2.1

Factor $2$ out of $2$ .

$- \frac{1}{4} - (\frac{2 \cdot 0}{2 \cdot 1} - \frac{{(- 1)}^{2}}{2})$

Step 3.7.2.3.7.2.2

Cancel the common factor.

$- \frac{1}{4} - (\frac{2 \cdot 0}{2 \cdot 1} - \frac{{(- 1)}^{2}}{2})$

Step 3.7.2.3.7.2.3

Rewrite the expression.

$- \frac{1}{4} - (\frac{0}{1} - \frac{{(- 1)}^{2}}{2})$

Step 3.7.2.3.7.2.4

Divide $0$ by $1$ .

$- \frac{1}{4} - (0 - \frac{{(- 1)}^{2}}{2})$

Step 3.7.2.3.8

Raise $- 1$ to the power of $2$ .

$- \frac{1}{4} - (0 - \frac{1}{2})$

Step 3.7.2.3.9

Subtract $\frac{1}{2}$ from $0$ .

$- \frac{1}{4} - - \frac{1}{2}$

Step 3.7.2.3.10

Multiply $- 1$ by $- 1$ .

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

Step 3.7.2.3.11

Multiply $\frac{1}{2}$ by $1$ .

$- \frac{1}{4} + \frac{1}{2}$

Step 3.7.2.3.12

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{1}{4} + \frac{1}{2} \cdot \frac{2}{2}$

Step 3.7.2.3.13

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 3.7.2.3.13.1

Multiply $\frac{1}{2}$ by $\frac{2}{2}$ .

$- \frac{1}{4} + \frac{2}{2 \cdot 2}$

Step 3.7.2.3.13.2

Multiply $2$ by $2$ .

$- \frac{1}{4} + \frac{2}{4}$

Step 3.7.2.3.14

Combine the numerators over the common denominator.

$\frac{- 1 + 2}{4}$

Step 3.7.2.3.15

Add $- 1$ and $2$ .

$\frac{1}{4}$

Step 4

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

Step 5

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

Tap for more steps...

Step 5.1

Combine the integrals into a single integral.

$\int_{0}^{1} x - (x^{3}) d x$

Step 5.2

Multiply $- 1$ by $x^{3}$ .

$\int_{0}^{1} x - x^{3} d x$

Step 5.3

Split the single integral into multiple integrals.

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

Step 5.4

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

$\frac{1}{2} x^{2}]_{0}^{1} + \int_{0}^{1} - x^{3} d x$

Step 5.5

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

$\frac{1}{2} x^{2}]_{0}^{1} - \int_{0}^{1} x^{3} d x$

Step 5.6

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

$\frac{1}{2} x^{2}]_{0}^{1} - (\frac{1}{4} x^{4}]_{0}^{1})$

Step 5.7

Simplify the answer.

Tap for more steps...

Step 5.7.1

Combine $\frac{1}{4}$ and $x^{4}$ .

$\frac{1}{2} x^{2}]_{0}^{1} - (\frac{x^{4}}{4}]_{0}^{1})$

Step 5.7.2

Substitute and simplify.

Tap for more steps...

Step 5.7.2.1

Evaluate $\frac{1}{2} x^{2}$ at $1$ and at $0$ .

$(\frac{1}{2} \cdot 1^{2}) - \frac{1}{2} \cdot 0^{2} - (\frac{x^{4}}{4}]_{0}^{1})$

Step 5.7.2.2

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

$\frac{1}{2} \cdot 1^{2} - \frac{1}{2} \cdot 0^{2} - (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 5.7.2.3

Simplify.

Tap for more steps...

Step 5.7.2.3.1

One to any power is one.

$\frac{1}{2} \cdot 1 - \frac{1}{2} \cdot 0^{2} - (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 5.7.2.3.2

Multiply $\frac{1}{2}$ by $1$ .

$\frac{1}{2} - \frac{1}{2} \cdot 0^{2} - (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 5.7.2.3.3

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

$\frac{1}{2} - \frac{1}{2} \cdot 0 - (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 5.7.2.3.4

Multiply $0$ by $- 1$ .

$\frac{1}{2} + 0 (\frac{1}{2}) - (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 5.7.2.3.5

Multiply $0$ by $\frac{1}{2}$ .

$\frac{1}{2} + 0 - (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 5.7.2.3.6

Add $\frac{1}{2}$ and $0$ .

$\frac{1}{2} - (\frac{1^{4}}{4} - \frac{0^{4}}{4})$

Step 5.7.2.3.7

One to any power is one.

$\frac{1}{2} - (\frac{1}{4} - \frac{0^{4}}{4})$

Step 5.7.2.3.8

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

$\frac{1}{2} - (\frac{1}{4} - \frac{0}{4})$

Step 5.7.2.3.9

Cancel the common factor of $0$ and $4$ .

Tap for more steps...

Step 5.7.2.3.9.1

Factor $4$ out of $0$ .

$\frac{1}{2} - (\frac{1}{4} - \frac{4 (0)}{4})$

Step 5.7.2.3.9.2

Cancel the common factors.

Tap for more steps...

Step 5.7.2.3.9.2.1

Factor $4$ out of $4$ .

$\frac{1}{2} - (\frac{1}{4} - \frac{4 \cdot 0}{4 \cdot 1})$

Step 5.7.2.3.9.2.2

Cancel the common factor.

$\frac{1}{2} - (\frac{1}{4} - \frac{4 \cdot 0}{4 \cdot 1})$

Step 5.7.2.3.9.2.3

Rewrite the expression.

$\frac{1}{2} - (\frac{1}{4} - \frac{0}{1})$

Step 5.7.2.3.9.2.4

Divide $0$ by $1$ .

$\frac{1}{2} - (\frac{1}{4} - 0)$

Step 5.7.2.3.10

Multiply $- 1$ by $0$ .

$\frac{1}{2} - (\frac{1}{4} + 0)$

Step 5.7.2.3.11

Add $\frac{1}{4}$ and $0$ .

$\frac{1}{2} - \frac{1}{4}$

Step 5.7.2.3.12

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{1}{2} \cdot \frac{2}{2} - \frac{1}{4}$

Step 5.7.2.3.13

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 5.7.2.3.13.1

Multiply $\frac{1}{2}$ by $\frac{2}{2}$ .

$\frac{2}{2 \cdot 2} - \frac{1}{4}$

Step 5.7.2.3.13.2

Multiply $2$ by $2$ .

$\frac{2}{4} - \frac{1}{4}$

Step 5.7.2.3.14

Combine the numerators over the common denominator.

$\frac{2 - 1}{4}$

Step 5.7.2.3.15

Subtract $1$ from $2$ .

$\frac{1}{4}$

Step 6

Add the areas $\frac{1}{4} + \frac{1}{4}$ .

Tap for more steps...

Step 6.1

Combine the numerators over the common denominator.

$\frac{1 + 1}{4}$

Step 6.2

Add $1$ and $1$ .

$\frac{2}{4}$

Step 6.3

Cancel the common factor of $2$ and $4$ .

Tap for more steps...

Step 6.3.1

Factor $2$ out of $2$ .

$\frac{2 (1)}{4}$

Step 6.3.2

Cancel the common factors.

Tap for more steps...

Step 6.3.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 1}{2 \cdot 2}$

Step 6.3.2.2

Cancel the common factor.

$\frac{2 \cdot 1}{2 \cdot 2}$

Step 6.3.2.3

Rewrite the expression.

$\frac{1}{2}$

Step 7