Calculus Examples

$y = 3 x - x^{2}$ , $y = x^{2}$

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.

$3 x - x^{2} = x^{2}$

Step 1.2

Solve $3 x - x^{2} = x^{2}$ 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 $x^{2}$ from both sides of the equation.

$3 x - x^{2} - x^{2} = 0$

Step 1.2.1.2

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

$3 x - 2 x^{2} = 0$

Step 1.2.2

Factor the left side of the equation.

Tap for more steps...

Step 1.2.2.1

Let $u = x$ . Substitute $u$ for all occurrences of $x$ .

$3 u - 2 u^{2} = 0$

Step 1.2.2.2

Factor $u$ out of $3 u - 2 u^{2}$ .

Tap for more steps...

Step 1.2.2.2.1

Factor $u$ out of $3 u$ .

$u \cdot 3 - 2 u^{2} = 0$

Step 1.2.2.2.2

Factor $u$ out of $- 2 u^{2}$ .

$u \cdot 3 + u (- 2 u) = 0$

Step 1.2.2.2.3

Factor $u$ out of $u \cdot 3 + u (- 2 u)$ .

$u (3 - 2 u) = 0$

Step 1.2.2.3

Replace all occurrences of $u$ with $x$ .

$x (3 - 2 x) = 0$

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 $0$ .

$x = 0$

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

Step 1.2.4

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.5

Set $3 - 2 x$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 1.2.5.1

Set $3 - 2 x$ equal to $0$ .

$3 - 2 x = 0$

Step 1.2.5.2

Solve $3 - 2 x = 0$ for $x$ .

Tap for more steps...

Step 1.2.5.2.1

Subtract $3$ from both sides of the equation.

$- 2 x = - 3$

Step 1.2.5.2.2

Divide each term in $- 2 x = - 3$ by $- 2$ and simplify.

Tap for more steps...

Step 1.2.5.2.2.1

Divide each term in $- 2 x = - 3$ by $- 2$ .

$\frac{- 2 x}{- 2} = \frac{- 3}{- 2}$

Step 1.2.5.2.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.5.2.2.2.1

Cancel the common factor of $- 2$ .

Tap for more steps...

Step 1.2.5.2.2.2.1.1

Cancel the common factor.

$\frac{- 2 x}{- 2} = \frac{- 3}{- 2}$

Step 1.2.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{- 2}$

Step 1.2.5.2.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.5.2.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{3}{2}$

Step 1.2.6

The final solution is all the values that make $x (3 - 2 x) = 0$ true.

$x = 0, \frac{3}{2}$

Step 1.3

Evaluate $y$ when $x = 0$ .

Tap for more steps...

Step 1.3.1

Substitute $0$ for $x$ .

$y = {(0)}^{2}$

Step 1.3.2

Substitute $0$ for $x$ in $y = {(0)}^{2}$ and solve for $y$ .

Tap for more steps...

Step 1.3.2.1

Remove parentheses.

$y = 0^{2}$

Step 1.3.2.2

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

$y = 0$

Step 1.4

Evaluate $y$ when $x = \frac{3}{2}$ .

Tap for more steps...

Step 1.4.1

Substitute $\frac{3}{2}$ for $x$ .

$y = {(\frac{3}{2})}^{2}$

Step 1.4.2

Simplify ${(\frac{3}{2})}^{2}$ .

Tap for more steps...

Step 1.4.2.1

Apply the product rule to $\frac{3}{2}$ .

$y = \frac{3^{2}}{2^{2}}$

Step 1.4.2.2

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

$y = \frac{9}{2^{2}}$

Step 1.4.2.3

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

$y = \frac{9}{4}$

Step 1.5

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

$(0, 0)$

$(\frac{3}{2}, \frac{9}{4})$

$(0, 0)$

$(\frac{3}{2}, \frac{9}{4})$

Step 2

Reorder $3 x$ and $- x^{2}$ .

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

Step 3

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.

$A r e a = \int_{0}^{\frac{3}{2}} - x^{2} + 3 x d x - \int_{0}^{\frac{3}{2}} x^{2} d x$

Step 4

Integrate to find the area between $0$ and $\frac{3}{2}$ .

Tap for more steps...

Step 4.1

Combine the integrals into a single integral.

$\int_{0}^{\frac{3}{2}} - x^{2} + 3 x - (x^{2}) d x$

Step 4.2

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

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

Step 4.3

Split the single integral into multiple integrals.

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

Step 4.4

Since $- 2$ is constant with respect to $x$ , move $- 2$ out of the integral.

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

Step 4.5

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

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

Step 4.6

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

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

Step 4.7

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

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

Step 4.8

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

$- 2 (\frac{x^{3}}{3}]_{0}^{\frac{3}{2}}) + 3 (\frac{1}{2} x^{2}]_{0}^{\frac{3}{2}})$

Step 4.9

Simplify the answer.

Tap for more steps...

Step 4.9.1

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

$- 2 (\frac{x^{3}}{3}]_{0}^{\frac{3}{2}}) + 3 (\frac{x^{2}}{2}]_{0}^{\frac{3}{2}})$

Step 4.9.2

Substitute and simplify.

Tap for more steps...

Step 4.9.2.1

Evaluate $\frac{x^{3}}{3}$ at $\frac{3}{2}$ and at $0$ .

$- 2 ((\frac{{(\frac{3}{2})}^{3}}{3}) - \frac{0^{3}}{3}) + 3 (\frac{x^{2}}{2}]_{0}^{\frac{3}{2}})$

Step 4.9.2.2

Evaluate $\frac{x^{2}}{2}$ at $\frac{3}{2}$ and at $0$ .

$- 2 (\frac{{(\frac{3}{2})}^{3}}{3} - \frac{0^{3}}{3}) + 3 (\frac{{(\frac{3}{2})}^{2}}{2} - \frac{0^{2}}{2})$

Step 4.9.2.3

Simplify.

Tap for more steps...

Step 4.9.2.3.1

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

$- 2 (\frac{{(\frac{3}{2})}^{3}}{3} - \frac{0}{3}) + 3 (\frac{{(\frac{3}{2})}^{2}}{2} - \frac{0^{2}}{2})$

Step 4.9.2.3.2

Cancel the common factor of $0$ and $3$ .

Tap for more steps...

Step 4.9.2.3.2.1

Factor $3$ out of $0$ .

$- 2 (\frac{{(\frac{3}{2})}^{3}}{3} - \frac{3 (0)}{3}) + 3 (\frac{{(\frac{3}{2})}^{2}}{2} - \frac{0^{2}}{2})$

Step 4.9.2.3.2.2

Cancel the common factors.

Tap for more steps...

Step 4.9.2.3.2.2.1

Factor $3$ out of $3$ .

$- 2 (\frac{{(\frac{3}{2})}^{3}}{3} - \frac{3 \cdot 0}{3 \cdot 1}) + 3 (\frac{{(\frac{3}{2})}^{2}}{2} - \frac{0^{2}}{2})$

Step 4.9.2.3.2.2.2

Cancel the common factor.

$- 2 (\frac{{(\frac{3}{2})}^{3}}{3} - \frac{3 \cdot 0}{3 \cdot 1}) + 3 (\frac{{(\frac{3}{2})}^{2}}{2} - \frac{0^{2}}{2})$

Step 4.9.2.3.2.2.3

Rewrite the expression.

$- 2 (\frac{{(\frac{3}{2})}^{3}}{3} - \frac{0}{1}) + 3 (\frac{{(\frac{3}{2})}^{2}}{2} - \frac{0^{2}}{2})$

Step 4.9.2.3.2.2.4

Divide $0$ by $1$ .

$- 2 (\frac{{(\frac{3}{2})}^{3}}{3} - 0) + 3 (\frac{{(\frac{3}{2})}^{2}}{2} - \frac{0^{2}}{2})$

Step 4.9.2.3.3

Multiply $- 1$ by $0$ .

$- 2 (\frac{{(\frac{3}{2})}^{3}}{3} + 0) + 3 (\frac{{(\frac{3}{2})}^{2}}{2} - \frac{0^{2}}{2})$

Step 4.9.2.3.4

Add $\frac{{(\frac{3}{2})}^{3}}{3}$ and $0$ .

$- 2 \frac{{(\frac{3}{2})}^{3}}{3} + 3 (\frac{{(\frac{3}{2})}^{2}}{2} - \frac{0^{2}}{2})$

Step 4.9.2.3.5

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

$- 2 \frac{{(\frac{3}{2})}^{3}}{3} + 3 (\frac{{(\frac{3}{2})}^{2}}{2} - \frac{0}{2})$

Step 4.9.2.3.6

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

Tap for more steps...

Step 4.9.2.3.6.1

Factor $2$ out of $0$ .

$- 2 \frac{{(\frac{3}{2})}^{3}}{3} + 3 (\frac{{(\frac{3}{2})}^{2}}{2} - \frac{2 (0)}{2})$

Step 4.9.2.3.6.2

Cancel the common factors.

Tap for more steps...

Step 4.9.2.3.6.2.1

Factor $2$ out of $2$ .

$- 2 \frac{{(\frac{3}{2})}^{3}}{3} + 3 (\frac{{(\frac{3}{2})}^{2}}{2} - \frac{2 \cdot 0}{2 \cdot 1})$

Step 4.9.2.3.6.2.2

Cancel the common factor.

$- 2 \frac{{(\frac{3}{2})}^{3}}{3} + 3 (\frac{{(\frac{3}{2})}^{2}}{2} - \frac{2 \cdot 0}{2 \cdot 1})$

Step 4.9.2.3.6.2.3

Rewrite the expression.

$- 2 \frac{{(\frac{3}{2})}^{3}}{3} + 3 (\frac{{(\frac{3}{2})}^{2}}{2} - \frac{0}{1})$

Step 4.9.2.3.6.2.4

Divide $0$ by $1$ .

$- 2 \frac{{(\frac{3}{2})}^{3}}{3} + 3 (\frac{{(\frac{3}{2})}^{2}}{2} - 0)$

Step 4.9.2.3.7

Multiply $- 1$ by $0$ .

$- 2 \frac{{(\frac{3}{2})}^{3}}{3} + 3 (\frac{{(\frac{3}{2})}^{2}}{2} + 0)$

Step 4.9.2.3.8

Add $\frac{{(\frac{3}{2})}^{2}}{2}$ and $0$ .

$- 2 \frac{{(\frac{3}{2})}^{3}}{3} + 3 \frac{{(\frac{3}{2})}^{2}}{2}$

Step 4.9.3

Simplify.

Tap for more steps...

Step 4.9.3.1

Simplify each term.

Tap for more steps...

Step 4.9.3.1.1

Simplify the numerator.

Tap for more steps...

Step 4.9.3.1.1.1

Apply the product rule to $\frac{3}{2}$ .

$- 2 \frac{\frac{3^{3}}{2^{3}}}{3} + 3 \frac{{(\frac{3}{2})}^{2}}{2}$

Step 4.9.3.1.1.2

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

$- 2 \frac{\frac{27}{2^{3}}}{3} + 3 \frac{{(\frac{3}{2})}^{2}}{2}$

Step 4.9.3.1.1.3

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

$- 2 \frac{\frac{27}{8}}{3} + 3 \frac{{(\frac{3}{2})}^{2}}{2}$

Step 4.9.3.1.2

Multiply the numerator by the reciprocal of the denominator.

$- 2 (\frac{27}{8} \cdot \frac{1}{3}) + 3 \frac{{(\frac{3}{2})}^{2}}{2}$

Step 4.9.3.1.3

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.9.3.1.3.1

Factor $3$ out of $27$ .

$- 2 (\frac{3 (9)}{8} \cdot \frac{1}{3}) + 3 \frac{{(\frac{3}{2})}^{2}}{2}$

Step 4.9.3.1.3.2

Cancel the common factor.

$- 2 (\frac{3 \cdot 9}{8} \cdot \frac{1}{3}) + 3 \frac{{(\frac{3}{2})}^{2}}{2}$

Step 4.9.3.1.3.3

Rewrite the expression.

$- 2 (\frac{9}{8}) + 3 \frac{{(\frac{3}{2})}^{2}}{2}$

Step 4.9.3.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 4.9.3.1.4.1

Factor $2$ out of $- 2$ .

$2 (- 1) \frac{9}{8} + 3 \frac{{(\frac{3}{2})}^{2}}{2}$

Step 4.9.3.1.4.2

Factor $2$ out of $8$ .

$2 \cdot - 1 \frac{9}{2 \cdot 4} + 3 \frac{{(\frac{3}{2})}^{2}}{2}$

Step 4.9.3.1.4.3

Cancel the common factor.

$2 \cdot - 1 \frac{9}{2 \cdot 4} + 3 \frac{{(\frac{3}{2})}^{2}}{2}$

Step 4.9.3.1.4.4

Rewrite the expression.

$- \frac{9}{4} + 3 \frac{{(\frac{3}{2})}^{2}}{2}$

Step 4.9.3.1.5

Simplify the numerator.

Tap for more steps...

Step 4.9.3.1.5.1

Apply the product rule to $\frac{3}{2}$ .

$- \frac{9}{4} + 3 \frac{\frac{3^{2}}{2^{2}}}{2}$

Step 4.9.3.1.5.2

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

$- \frac{9}{4} + 3 \frac{\frac{9}{2^{2}}}{2}$

Step 4.9.3.1.5.3

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

$- \frac{9}{4} + 3 \frac{\frac{9}{4}}{2}$

Step 4.9.3.1.6

Multiply the numerator by the reciprocal of the denominator.

$- \frac{9}{4} + 3 (\frac{9}{4} \cdot \frac{1}{2})$

Step 4.9.3.1.7

Multiply $\frac{9}{4} \cdot \frac{1}{2}$ .

Tap for more steps...

Step 4.9.3.1.7.1

Multiply $\frac{9}{4}$ by $\frac{1}{2}$ .

$- \frac{9}{4} + 3 \frac{9}{4 \cdot 2}$

Step 4.9.3.1.7.2

Multiply $4$ by $2$ .

$- \frac{9}{4} + 3 (\frac{9}{8})$

Step 4.9.3.1.8

Multiply $3 (\frac{9}{8})$ .

Tap for more steps...

Step 4.9.3.1.8.1

Combine $3$ and $\frac{9}{8}$ .

$- \frac{9}{4} + \frac{3 \cdot 9}{8}$

Step 4.9.3.1.8.2

Multiply $3$ by $9$ .

$- \frac{9}{4} + \frac{27}{8}$

Step 4.9.3.2

To write $- \frac{9}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- \frac{9}{4} \cdot \frac{2}{2} + \frac{27}{8}$

Step 4.9.3.3

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 4.9.3.3.1

Multiply $\frac{9}{4}$ by $\frac{2}{2}$ .

$- \frac{9 \cdot 2}{4 \cdot 2} + \frac{27}{8}$

Step 4.9.3.3.2

Multiply $4$ by $2$ .

$- \frac{9 \cdot 2}{8} + \frac{27}{8}$

Step 4.9.3.4

Combine the numerators over the common denominator.

$\frac{- 9 \cdot 2 + 27}{8}$

Step 4.9.3.5

Simplify the numerator.

Tap for more steps...

Step 4.9.3.5.1

Multiply $- 9$ by $2$ .

$\frac{- 18 + 27}{8}$

Step 4.9.3.5.2

Add $- 18$ and $27$ .

$\frac{9}{8}$

Step 5

Enter YOUR Problem