Calculus Examples

$y = x^{3} - 15 x$ , $y = x$

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.

$x^{3} - 15 x = x$

Step 1.2

Solve $x^{3} - 15 x = x$ for $x$ .

Tap for more steps...

Step 1.2.1

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

Tap for more steps...

Step 1.2.1.1

Subtract $x$ from both sides of the equation.

$x^{3} - 15 x - x = 0$

Step 1.2.1.2

Subtract $x$ from $- 15 x$ .

$x^{3} - 16 x = 0$

Step 1.2.2

Factor the left side of the equation.

Tap for more steps...

Step 1.2.2.1

Factor $x$ out of $x^{3} - 16 x$ .

Tap for more steps...

Step 1.2.2.1.1

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

$x \cdot x^{2} - 16 x = 0$

Step 1.2.2.1.2

Factor $x$ out of $- 16 x$ .

$x \cdot x^{2} + x \cdot - 16 = 0$

Step 1.2.2.1.3

Factor $x$ out of $x \cdot x^{2} + x \cdot - 16$ .

$x (x^{2} - 16) = 0$

Step 1.2.2.2

Rewrite $16$ as $4^{2}$ .

$x (x^{2} - 4^{2}) = 0$

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 $a = x$ and $b = 4$ .

$x ((x + 4) (x - 4)) = 0$

Step 1.2.2.3.2

Remove unnecessary parentheses.

$x (x + 4) (x - 4) = 0$

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 $0$ .

$x = 0$

$x + 4 = 0$

$x - 4 = 0 + y = x$

Step 1.2.4

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.5

Set $x + 4$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 1.2.5.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 1.2.5.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 1.2.6

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

Tap for more steps...

Step 1.2.6.1

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

$x - 4 = 0$

Step 1.2.6.2

Add $4$ to both sides of the equation.

$x = 4$

Step 1.2.7

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

$x = 0, - 4, 4$

Step 1.3

Evaluate $y$ when $x = 0$ .

Tap for more steps...

Step 1.3.1

Substitute $0$ for $x$ .

$y = 0$

Step 1.3.2

Remove parentheses.

$y = 0$

Step 1.4

Evaluate $y$ when $x = - 4$ .

Tap for more steps...

Step 1.4.1

Substitute $- 4$ for $x$ .

$y = - 4$

Step 1.4.2

Remove parentheses.

$y = - 4$

Step 1.5

Evaluate $y$ when $x = 4$ .

Tap for more steps...

Step 1.5.1

Substitute $4$ for $x$ .

$y = 4$

Step 1.5.2

Remove parentheses.

$y = 4$

Step 1.6

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

$(0, 0)$

$(- 4, - 4)$

$(4, 4)$

$(0, 0)$

$(- 4, - 4)$

$(4, 4)$

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.

$A r e a = \int_{- 4}^{0} x^{3} - 15 x d x - \int_{- 4}^{0} x d x$

Step 3

Integrate to find the area between $- 4$ and $0$ .

Tap for more steps...

Step 3.1

Combine the integrals into a single integral.

$\int_{- 4}^{0} x^{3} - 15 x - (x) d x$

Step 3.2

Subtract $x$ from $- 15 x$ .

$\int_{- 4}^{0} x^{3} - 16 x d x$

Step 3.3

Split the single integral into multiple integrals.

$\int_{- 4}^{0} x^{3} d x + \int_{- 4}^{0} - 16 x d x$

Step 3.4

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

$\frac{1}{4} x^{4}]_{- 4}^{0} + \int_{- 4}^{0} - 16 x d x$

Step 3.5

Since $- 16$ is constant with respect to $x$ , move $- 16$ out of the integral.

$\frac{1}{4} x^{4}]_{- 4}^{0} - 16 \int_{- 4}^{0} x d x$

Step 3.6

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

$\frac{1}{4} x^{4}]_{- 4}^{0} - 16 (\frac{1}{2} x^{2}]_{- 4}^{0})$

Step 3.7

Simplify the answer.

Tap for more steps...

Step 3.7.1

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

$\frac{1}{4} x^{4}]_{- 4}^{0} - 16 (\frac{x^{2}}{2}]_{- 4}^{0})$

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 $- 4$ .

$(\frac{1}{4} \cdot 0^{4}) - \frac{1}{4} {(- 4)}^{4} - 16 (\frac{x^{2}}{2}]_{- 4}^{0})$

Step 3.7.2.2

Evaluate $\frac{x^{2}}{2}$ at $0$ and at $- 4$ .

$\frac{1}{4} \cdot 0^{4} - \frac{1}{4} {(- 4)}^{4} - 16 (\frac{0^{2}}{2} - \frac{{(- 4)}^{2}}{2})$

Step 3.7.2.3

Simplify.

Tap for more steps...

Step 3.7.2.3.1

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

$\frac{1}{4} \cdot 0 - \frac{1}{4} {(- 4)}^{4} - 16 (\frac{0^{2}}{2} - \frac{{(- 4)}^{2}}{2})$

Step 3.7.2.3.2

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

$0 - \frac{1}{4} {(- 4)}^{4} - 16 (\frac{0^{2}}{2} - \frac{{(- 4)}^{2}}{2})$

Step 3.7.2.3.3

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

$0 - \frac{1}{4} \cdot 256 - 16 (\frac{0^{2}}{2} - \frac{{(- 4)}^{2}}{2})$

Step 3.7.2.3.4

Multiply $256$ by $- 1$ .

$0 - 256 (\frac{1}{4}) - 16 (\frac{0^{2}}{2} - \frac{{(- 4)}^{2}}{2})$

Step 3.7.2.3.5

Combine $- 256$ and $\frac{1}{4}$ .

$0 + \frac{- 256}{4} - 16 (\frac{0^{2}}{2} - \frac{{(- 4)}^{2}}{2})$

Step 3.7.2.3.6

Cancel the common factor of $- 256$ and $4$ .

Tap for more steps...

Step 3.7.2.3.6.1

Factor $4$ out of $- 256$ .

$0 + \frac{4 \cdot - 64}{4} - 16 (\frac{0^{2}}{2} - \frac{{(- 4)}^{2}}{2})$

Step 3.7.2.3.6.2

Cancel the common factors.

Tap for more steps...

Step 3.7.2.3.6.2.1

Factor $4$ out of $4$ .

$0 + \frac{4 \cdot - 64}{4 (1)} - 16 (\frac{0^{2}}{2} - \frac{{(- 4)}^{2}}{2})$

Step 3.7.2.3.6.2.2

Cancel the common factor.

$0 + \frac{4 \cdot - 64}{4 \cdot 1} - 16 (\frac{0^{2}}{2} - \frac{{(- 4)}^{2}}{2})$

Step 3.7.2.3.6.2.3

Rewrite the expression.

$0 + \frac{- 64}{1} - 16 (\frac{0^{2}}{2} - \frac{{(- 4)}^{2}}{2})$

Step 3.7.2.3.6.2.4

Divide $- 64$ by $1$ .

$0 - 64 - 16 (\frac{0^{2}}{2} - \frac{{(- 4)}^{2}}{2})$

Step 3.7.2.3.7

Subtract $64$ from $0$ .

$- 64 - 16 (\frac{0^{2}}{2} - \frac{{(- 4)}^{2}}{2})$

Step 3.7.2.3.8

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

$- 64 - 16 (\frac{0}{2} - \frac{{(- 4)}^{2}}{2})$

Step 3.7.2.3.9

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

Tap for more steps...

Step 3.7.2.3.9.1

Factor $2$ out of $0$ .

$- 64 - 16 (\frac{2 (0)}{2} - \frac{{(- 4)}^{2}}{2})$

Step 3.7.2.3.9.2

Cancel the common factors.

Tap for more steps...

Step 3.7.2.3.9.2.1

Factor $2$ out of $2$ .

$- 64 - 16 (\frac{2 \cdot 0}{2 \cdot 1} - \frac{{(- 4)}^{2}}{2})$

Step 3.7.2.3.9.2.2

Cancel the common factor.

$- 64 - 16 (\frac{2 \cdot 0}{2 \cdot 1} - \frac{{(- 4)}^{2}}{2})$

Step 3.7.2.3.9.2.3

Rewrite the expression.

$- 64 - 16 (\frac{0}{1} - \frac{{(- 4)}^{2}}{2})$

Step 3.7.2.3.9.2.4

Divide $0$ by $1$ .

$- 64 - 16 (0 - \frac{{(- 4)}^{2}}{2})$

Step 3.7.2.3.10

Raise $- 4$ to the power of $2$ .

$- 64 - 16 (0 - \frac{16}{2})$

Step 3.7.2.3.11

Cancel the common factor of $16$ and $2$ .

Tap for more steps...

Step 3.7.2.3.11.1

Factor $2$ out of $16$ .

$- 64 - 16 (0 - \frac{2 \cdot 8}{2})$

Step 3.7.2.3.11.2

Cancel the common factors.

Tap for more steps...

Step 3.7.2.3.11.2.1

Factor $2$ out of $2$ .

$- 64 - 16 (0 - \frac{2 \cdot 8}{2 (1)})$

Step 3.7.2.3.11.2.2

Cancel the common factor.

$- 64 - 16 (0 - \frac{2 \cdot 8}{2 \cdot 1})$

Step 3.7.2.3.11.2.3

Rewrite the expression.

$- 64 - 16 (0 - \frac{8}{1})$

Step 3.7.2.3.11.2.4

Divide $8$ by $1$ .

$- 64 - 16 (0 - 1 \cdot 8)$

Step 3.7.2.3.12

Multiply $- 1$ by $8$ .

$- 64 - 16 (0 - 8)$

Step 3.7.2.3.13

Subtract $8$ from $0$ .

$- 64 - 16 \cdot - 8$

Step 3.7.2.3.14

Multiply $- 16$ by $- 8$ .

$- 64 + 128$

Step 3.7.2.3.15

Add $- 64$ and $128$ .

$64$

Step 4

$A r e a = \int_{0}^{4} x d x - \int_{0}^{4} x^{3} - 15 x d x$

Step 5

Integrate to find the area between $0$ and $4$ .

Tap for more steps...

Step 5.1

Combine the integrals into a single integral.

$\int_{0}^{4} x - (x^{3} - 15 x) d x$

Step 5.2

Simplify each term.

Tap for more steps...

Step 5.2.1

Apply the distributive property.

$x - x^{3} - (- 15 x)$

Step 5.2.2

Multiply $- 15$ by $- 1$ .

$x - x^{3} + 15 x$

$\int_{0}^{4} x - x^{3} + 15 x d x$

Step 5.3

Add $x$ and $15 x$ .

$\int_{0}^{4} - x^{3} + 16 x d x$

Step 5.4

Split the single integral into multiple integrals.

$\int_{0}^{4} - x^{3} d x + \int_{0}^{4} 16 x d x$

Step 5.5

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

$- \int_{0}^{4} x^{3} d x + \int_{0}^{4} 16 x d x$

Step 5.6

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

$- (\frac{1}{4} x^{4}]_{0}^{4}) + \int_{0}^{4} 16 x d x$

Step 5.7

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

$- (\frac{x^{4}}{4}]_{0}^{4}) + \int_{0}^{4} 16 x d x$

Step 5.8

Since $16$ is constant with respect to $x$ , move $16$ out of the integral.

$- (\frac{x^{4}}{4}]_{0}^{4}) + 16 \int_{0}^{4} x d x$

Step 5.9

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

$- (\frac{x^{4}}{4}]_{0}^{4}) + 16 (\frac{1}{2} x^{2}]_{0}^{4})$

Step 5.10

Simplify the answer.

Tap for more steps...

Step 5.10.1

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

$- (\frac{x^{4}}{4}]_{0}^{4}) + 16 (\frac{x^{2}}{2}]_{0}^{4})$

Step 5.10.2

Substitute and simplify.

Tap for more steps...

Step 5.10.2.1

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

$- ((\frac{4^{4}}{4}) - \frac{0^{4}}{4}) + 16 (\frac{x^{2}}{2}]_{0}^{4})$

Step 5.10.2.2

Evaluate $\frac{x^{2}}{2}$ at $4$ and at $0$ .

$- (\frac{4^{4}}{4} - \frac{0^{4}}{4}) + 16 (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 5.10.2.3

Simplify.

Tap for more steps...

Step 5.10.2.3.1

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

$- (\frac{256}{4} - \frac{0^{4}}{4}) + 16 (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 5.10.2.3.2

Cancel the common factor of $256$ and $4$ .

Tap for more steps...

Step 5.10.2.3.2.1

Factor $4$ out of $256$ .

$- (\frac{4 \cdot 64}{4} - \frac{0^{4}}{4}) + 16 (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 5.10.2.3.2.2

Cancel the common factors.

Tap for more steps...

Step 5.10.2.3.2.2.1

Factor $4$ out of $4$ .

$- (\frac{4 \cdot 64}{4 (1)} - \frac{0^{4}}{4}) + 16 (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 5.10.2.3.2.2.2

Cancel the common factor.

$- (\frac{4 \cdot 64}{4 \cdot 1} - \frac{0^{4}}{4}) + 16 (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 5.10.2.3.2.2.3

Rewrite the expression.

$- (\frac{64}{1} - \frac{0^{4}}{4}) + 16 (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 5.10.2.3.2.2.4

Divide $64$ by $1$ .

$- (64 - \frac{0^{4}}{4}) + 16 (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 5.10.2.3.3

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

$- (64 - \frac{0}{4}) + 16 (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 5.10.2.3.4

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

Tap for more steps...

Step 5.10.2.3.4.1

Factor $4$ out of $0$ .

$- (64 - \frac{4 (0)}{4}) + 16 (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 5.10.2.3.4.2

Cancel the common factors.

Tap for more steps...

Step 5.10.2.3.4.2.1

Factor $4$ out of $4$ .

$- (64 - \frac{4 \cdot 0}{4 \cdot 1}) + 16 (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 5.10.2.3.4.2.2

Cancel the common factor.

$- (64 - \frac{4 \cdot 0}{4 \cdot 1}) + 16 (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 5.10.2.3.4.2.3

Rewrite the expression.

$- (64 - \frac{0}{1}) + 16 (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 5.10.2.3.4.2.4

Divide $0$ by $1$ .

$- (64 - 0) + 16 (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 5.10.2.3.5

Multiply $- 1$ by $0$ .

$- (64 + 0) + 16 (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 5.10.2.3.6

Add $64$ and $0$ .

$- 1 \cdot 64 + 16 (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 5.10.2.3.7

Multiply $- 1$ by $64$ .

$- 64 + 16 (\frac{4^{2}}{2} - \frac{0^{2}}{2})$

Step 5.10.2.3.8

Raise $4$ to the power of $2$ .

$- 64 + 16 (\frac{16}{2} - \frac{0^{2}}{2})$

Step 5.10.2.3.9

Cancel the common factor of $16$ and $2$ .

Tap for more steps...

Step 5.10.2.3.9.1

Factor $2$ out of $16$ .

$- 64 + 16 (\frac{2 \cdot 8}{2} - \frac{0^{2}}{2})$

Step 5.10.2.3.9.2

Cancel the common factors.

Tap for more steps...

Step 5.10.2.3.9.2.1

Factor $2$ out of $2$ .

$- 64 + 16 (\frac{2 \cdot 8}{2 (1)} - \frac{0^{2}}{2})$

Step 5.10.2.3.9.2.2

Cancel the common factor.

$- 64 + 16 (\frac{2 \cdot 8}{2 \cdot 1} - \frac{0^{2}}{2})$

Step 5.10.2.3.9.2.3

Rewrite the expression.

$- 64 + 16 (\frac{8}{1} - \frac{0^{2}}{2})$

Step 5.10.2.3.9.2.4

Divide $8$ by $1$ .

$- 64 + 16 (8 - \frac{0^{2}}{2})$

Step 5.10.2.3.10

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

$- 64 + 16 (8 - \frac{0}{2})$

Step 5.10.2.3.11

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

Tap for more steps...

Step 5.10.2.3.11.1

Factor $2$ out of $0$ .

$- 64 + 16 (8 - \frac{2 (0)}{2})$

Step 5.10.2.3.11.2

Cancel the common factors.

Tap for more steps...

Step 5.10.2.3.11.2.1

Factor $2$ out of $2$ .

$- 64 + 16 (8 - \frac{2 \cdot 0}{2 \cdot 1})$

Step 5.10.2.3.11.2.2

Cancel the common factor.

$- 64 + 16 (8 - \frac{2 \cdot 0}{2 \cdot 1})$

Step 5.10.2.3.11.2.3

Rewrite the expression.

$- 64 + 16 (8 - \frac{0}{1})$

Step 5.10.2.3.11.2.4

Divide $0$ by $1$ .

$- 64 + 16 (8 - 0)$

Step 5.10.2.3.12

Multiply $- 1$ by $0$ .

$- 64 + 16 (8 + 0)$

Step 5.10.2.3.13

Add $8$ and $0$ .

$- 64 + 16 \cdot 8$

Step 5.10.2.3.14

Multiply $16$ by $8$ .

$- 64 + 128$

Step 5.10.2.3.15

Add $- 64$ and $128$ .

$64$

Step 6

Add $64$ and $64$ .

$128$

Step 7