Calculus Examples

$y = \sqrt{3 - 7 x}$ , $x = 0$

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.

$\sqrt{3 - 7 x} = 0$

Step 1.2

Solve $\sqrt{3 - 7 x} = 0$ for $x$ .

Tap for more steps...

Step 1.2.1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{3 - 7 x}}^{2} = 0^{2}$

Step 1.2.2

Simplify each side of the equation.

Tap for more steps...

Step 1.2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 - 7 x}$ as ${(3 - 7 x)}^{\frac{1}{2}}$ .

${({(3 - 7 x)}^{\frac{1}{2}})}^{2} = 0^{2}$

Step 1.2.2.2

Simplify the left side.

Tap for more steps...

Step 1.2.2.2.1

Simplify ${({(3 - 7 x)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 1.2.2.2.1.1

Multiply the exponents in ${({(3 - 7 x)}^{\frac{1}{2}})}^{2}$ .

Tap for more steps...

Step 1.2.2.2.1.1.1

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

${(3 - 7 x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 1.2.2.2.1.1.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.2.2.2.1.1.2.1

Cancel the common factor.

${(3 - 7 x)}^{\frac{1}{2} \cdot 2} = 0^{2}$

Step 1.2.2.2.1.1.2.2

Rewrite the expression.

${(3 - 7 x)}^{1} = 0^{2}$

Step 1.2.2.2.1.2

Simplify.

$3 - 7 x = 0^{2}$

Step 1.2.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.2.3.1

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

$3 - 7 x = 0$

Step 1.2.3

Solve for $x$ .

Tap for more steps...

Step 1.2.3.1

Subtract $3$ from both sides of the equation.

$- 7 x = - 3$

Step 1.2.3.2

Divide each term in $- 7 x = - 3$ by $- 7$ and simplify.

Tap for more steps...

Step 1.2.3.2.1

Divide each term in $- 7 x = - 3$ by $- 7$ .

$\frac{- 7 x}{- 7} = \frac{- 3}{- 7}$

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 $- 7$ .

Tap for more steps...

Step 1.2.3.2.2.1.1

Cancel the common factor.

$\frac{- 7 x}{- 7} = \frac{- 3}{- 7}$

Step 1.2.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 3}{- 7}$

Step 1.2.3.2.3

Simplify the right side.

Tap for more steps...

Step 1.2.3.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{3}{7}$

Step 1.3

Substitute $\frac{3}{7}$ for $x$ .

$y = 0$

Step 1.4

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

$(\frac{3}{7}, 0)$

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.

$A r e a = \int_{0}^{\frac{3}{7}} \sqrt{3 - 7 x} d x - \int_{0}^{\frac{3}{7}} 0 d x$

Step 3

Integrate to find the area between $0$ and $\frac{3}{7}$ .

Tap for more steps...

Step 3.1

Combine the integrals into a single integral.

$\int_{0}^{\frac{3}{7}} \sqrt{3 - 7 x} - (0) d x$

Step 3.2

Subtract $0$ from $\sqrt{3 - 7 x}$ .

$\int_{0}^{\frac{3}{7}} \sqrt{3 - 7 x} d x$

Step 3.3

Let $u = 3 - 7 x$ . Then $d u = - 7 d x$ , so $- \frac{1}{7} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 3.3.1

Let $u = 3 - 7 x$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 3.3.1.1

Differentiate $3 - 7 x$ .

$\frac{d}{d x} [3 - 7 x]$

Step 3.3.1.2

Differentiate.

Tap for more steps...

Step 3.3.1.2.1

By the Sum Rule, the derivative of $3 - 7 x$ with respect to $x$ is $\frac{d}{d x} [3] + \frac{d}{d x} [- 7 x]$ .

$\frac{d}{d x} [3] + \frac{d}{d x} [- 7 x]$

Step 3.3.1.2.2

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- 7 x]$

Step 3.3.1.3

Evaluate $\frac{d}{d x} [- 7 x]$ .

Tap for more steps...

Step 3.3.1.3.1

Since $- 7$ is constant with respect to $x$ , the derivative of $- 7 x$ with respect to $x$ is $- 7 \frac{d}{d x} [x]$ .

$0 - 7 \frac{d}{d x} [x]$

Step 3.3.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$0 - 7 \cdot 1$

Step 3.3.1.3.3

Multiply $- 7$ by $1$ .

$0 - 7$

Step 3.3.1.4

Subtract $7$ from $0$ .

$- 7$

Step 3.3.2

Substitute the lower limit in for $x$ in $u = 3 - 7 x$ .

$u_{lower} = 3 - 7 \cdot 0$

Step 3.3.3

Simplify.

Tap for more steps...

Step 3.3.3.1

Multiply $- 7$ by $0$ .

$u_{lower} = 3 + 0$

Step 3.3.3.2

Add $3$ and $0$ .

$u_{lower} = 3$

Step 3.3.4

Substitute the upper limit in for $x$ in $u = 3 - 7 x$ .

$u_{upper} = 3 - 7 (\frac{3}{7})$

Step 3.3.5

Simplify.

Tap for more steps...

Step 3.3.5.1

Simplify each term.

Tap for more steps...

Step 3.3.5.1.1

Cancel the common factor of $7$ .

Tap for more steps...

Step 3.3.5.1.1.1

Factor $7$ out of $- 7$ .

$u_{upper} = 3 + 7 (- 1) \frac{3}{7}$

Step 3.3.5.1.1.2

Cancel the common factor.

$u_{upper} = 3 + 7 \cdot - 1 \frac{3}{7}$

Step 3.3.5.1.1.3

Rewrite the expression.

$u_{upper} = 3 - 1 \cdot 3$

Step 3.3.5.1.2

Multiply $- 1$ by $3$ .

$u_{upper} = 3 - 3$

Step 3.3.5.2

Subtract $3$ from $3$ .

$u_{upper} = 0$

Step 3.3.6

The values found for $u_{lower}$ and $u_{upper}$ will be used to evaluate the definite integral.

$u_{lower} = 3$

$u_{upper} = 0$

Step 3.3.7

Rewrite the problem using $u$ , $d u$ , and the new limits of integration.

$\int_{3}^{0} \sqrt{u} \frac{1}{- 7} d u$

Step 3.4

Simplify.

Tap for more steps...

Step 3.4.1

Move the negative in front of the fraction.

$\int_{3}^{0} \sqrt{u} (- \frac{1}{7}) d u$

Step 3.4.2

Combine $\sqrt{u}$ and $\frac{1}{7}$ .

$\int_{3}^{0} - \frac{\sqrt{u}}{7} d u$

Step 3.5

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

$- \int_{3}^{0} \frac{\sqrt{u}}{7} d u$

Step 3.6

Since $\frac{1}{7}$ is constant with respect to $u$ , move $\frac{1}{7}$ out of the integral.

$- (\frac{1}{7} \int_{3}^{0} \sqrt{u} d u)$

Step 3.7

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u}$ as $u^{\frac{1}{2}}$ .

$- \frac{1}{7} \int_{3}^{0} u^{\frac{1}{2}} d u$

Step 3.8

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

$- \frac{1}{7} \frac{2}{3} u^{\frac{3}{2}}]_{3}^{0}$

Step 3.9

Substitute and simplify.

Tap for more steps...

Step 3.9.1

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

$- \frac{1}{7} ((\frac{2}{3} \cdot 0^{\frac{3}{2}}) - \frac{2}{3} \cdot 3^{\frac{3}{2}})$

Step 3.9.2

Simplify.

Tap for more steps...

Step 3.9.2.1

Rewrite $0$ as $0^{2}$ .

$- \frac{1}{7} (\frac{2}{3} \cdot {(0^{2})}^{\frac{3}{2}} - \frac{2}{3} \cdot 3^{\frac{3}{2}})$

Step 3.9.2.2

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

$- \frac{1}{7} (\frac{2}{3} \cdot 0^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 3^{\frac{3}{2}})$

Step 3.9.2.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.9.2.3.1

Cancel the common factor.

$- \frac{1}{7} (\frac{2}{3} \cdot 0^{2 (\frac{3}{2})} - \frac{2}{3} \cdot 3^{\frac{3}{2}})$

Step 3.9.2.3.2

Rewrite the expression.

$- \frac{1}{7} (\frac{2}{3} \cdot 0^{3} - \frac{2}{3} \cdot 3^{\frac{3}{2}})$

Step 3.9.2.4

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

$- \frac{1}{7} (\frac{2}{3} \cdot 0 - \frac{2}{3} \cdot 3^{\frac{3}{2}})$

Step 3.9.2.5

Multiply $\frac{2}{3}$ by $0$ .

$- \frac{1}{7} (0 - \frac{2}{3} \cdot 3^{\frac{3}{2}})$

Step 3.9.2.6

Combine $3^{\frac{3}{2}}$ and $\frac{2}{3}$ .

$- \frac{1}{7} (0 - \frac{3^{\frac{3}{2}} \cdot 2}{3})$

Step 3.9.2.7

Move $2$ to the left of $3^{\frac{3}{2}}$ .

$- \frac{1}{7} (0 - \frac{2 \cdot 3^{\frac{3}{2}}}{3})$

Step 3.9.2.8

Move $3^{1}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$- \frac{1}{7} (0 - (2 \cdot 3^{\frac{3}{2}} \cdot 3^{- 1}))$

Step 3.9.2.9

Multiply $3^{\frac{3}{2}}$ by $3^{- 1}$ by adding the exponents.

Tap for more steps...

Step 3.9.2.9.1

Move $3^{- 1}$ .

$- \frac{1}{7} (0 - (2 \cdot (3^{- 1} \cdot 3^{\frac{3}{2}})))$

Step 3.9.2.9.2

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

$- \frac{1}{7} (0 - (2 \cdot 3^{- 1 + \frac{3}{2}}))$

Step 3.9.2.9.3

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

$- \frac{1}{7} (0 - (2 \cdot 3^{- 1 \cdot \frac{2}{2} + \frac{3}{2}}))$

Step 3.9.2.9.4

Combine $- 1$ and $\frac{2}{2}$ .

$- \frac{1}{7} (0 - (2 \cdot 3^{\frac{- 1 \cdot 2}{2} + \frac{3}{2}}))$

Step 3.9.2.9.5

Combine the numerators over the common denominator.

$- \frac{1}{7} (0 - (2 \cdot 3^{\frac{- 1 \cdot 2 + 3}{2}}))$

Step 3.9.2.9.6

Simplify the numerator.

Tap for more steps...

Step 3.9.2.9.6.1

Multiply $- 1$ by $2$ .

$- \frac{1}{7} (0 - (2 \cdot 3^{\frac{- 2 + 3}{2}}))$

Step 3.9.2.9.6.2

Add $- 2$ and $3$ .

$- \frac{1}{7} (0 - (2 \cdot 3^{\frac{1}{2}}))$

Step 3.9.2.10

Multiply $2$ by $- 1$ .

$- \frac{1}{7} (0 - 2 \cdot 3^{\frac{1}{2}})$

Step 3.9.2.11

Subtract $2 \cdot 3^{\frac{1}{2}}$ from $0$ .

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

Step 3.9.2.12

Multiply $- 2$ by $- 1$ .

$2 (\frac{1}{7}) \cdot 3^{\frac{1}{2}}$

Step 3.9.2.13

Combine $2$ and $\frac{1}{7}$ .

$\frac{2}{7} \cdot 3^{\frac{1}{2}}$

Step 3.9.2.14

Combine $\frac{2}{7}$ and $3^{\frac{1}{2}}$ .

$\frac{2 \cdot 3^{\frac{1}{2}}}{7}$

Step 4