Calculus Examples

$y = \sqrt{1 - x^{2}}$ , $y = 0$

Step 1

Solve by substitution to find the intersection between the curves.

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Step 1.1

Eliminate the equal sides of each equation and combine.

$\sqrt{1 - x^{2}} = 0$

Step 1.2

Solve $\sqrt{1 - x^{2}} = 0$ for $x$ .

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Step 1.2.1

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

${\sqrt{1 - x^{2}}}^{2} = 0^{2}$

Step 1.2.2

Simplify each side of the equation.

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Step 1.2.2.1

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

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

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

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

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Step 1.2.2.2.1.1

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

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Step 1.2.2.2.1.1.1

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

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

Step 1.2.2.2.1.1.2

Cancel the common factor of $2$ .

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Step 1.2.2.2.1.1.2.1

Cancel the common factor.

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

Step 1.2.2.2.1.1.2.2

Rewrite the expression.

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

Step 1.2.2.2.1.2

Simplify.

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

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

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

$1 - x^{2} = 0$

Step 1.2.3

Solve for $x$ .

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Step 1.2.3.1

Subtract $1$ from both sides of the equation.

$- x^{2} = - 1$

Step 1.2.3.2

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

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Step 1.2.3.2.1

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

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

Step 1.2.3.2.2

Simplify the left side.

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Step 1.2.3.2.2.1

Dividing two negative values results in a positive value.

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

Step 1.2.3.2.2.2

Divide $x^{2}$ by $1$ .

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

Step 1.2.3.2.3

Simplify the right side.

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Step 1.2.3.2.3.1

Divide $- 1$ by $- 1$ .

$x^{2} = 1$

Step 1.2.3.3

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

$x = \pm \sqrt{1}$

Step 1.2.3.4

Any root of $1$ is $1$ .

$x = \pm 1$

Step 1.2.3.5

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

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Step 1.2.3.5.1

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

$x = 1$

Step 1.2.3.5.2

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

$x = - 1$

Step 1.2.3.5.3

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

$x = 1, - 1$

Step 1.3

Substitute $1$ for $x$ .

$y = 0$

Step 1.4

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

$(1, 0)$

$(- 1, 0)$

$(1, 0)$

$(- 1, 0)$

Step 2

Simplify $\sqrt{1 - x^{2}}$ .

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Step 2.1

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

$y = \sqrt{1^{2} - x^{2}}$

Step 2.2

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 $b = x$ .

$y = \sqrt{(1 + x) (1 - x)}$

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.

$A r e a = \int_{- 1}^{1} \sqrt{(1 + x) (1 - x)} d x - \int_{- 1}^{1} 0 d x$

Step 4

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

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Step 4.1

Combine the integrals into a single integral.

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

Step 4.2

Subtract $0$ from $\sqrt{(1 + x) (1 - x)}$ .

$\int_{- 1}^{1} \sqrt{(1 + x) (1 - x)} d x$

Step 4.3

Complete the square.

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Step 4.3.1

Simplify the expression.

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Step 4.3.1.1

Expand $(1 + x) (1 - x)$ using the FOIL Method.

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Step 4.3.1.1.1

Apply the distributive property.

$1 (1 - x) + x (1 - x)$

Step 4.3.1.1.2

Apply the distributive property.

$1 \cdot 1 + 1 (- x) + x (1 - x)$

Step 4.3.1.1.3

Apply the distributive property.

$1 \cdot 1 + 1 (- x) + x \cdot 1 + x (- x)$

Step 4.3.1.2

Simplify and combine like terms.

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Step 4.3.1.2.1

Simplify each term.

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Step 4.3.1.2.1.1

Multiply $1$ by $1$ .

$1 + 1 (- x) + x \cdot 1 + x (- x)$

Step 4.3.1.2.1.2

Multiply $- x$ by $1$ .

$1 - x + x \cdot 1 + x (- x)$

Step 4.3.1.2.1.3

Multiply $x$ by $1$ .

$1 - x + x + x (- x)$

Step 4.3.1.2.1.4

Rewrite using the commutative property of multiplication.

$1 - x + x - x \cdot x$

Step 4.3.1.2.1.5

Multiply $x$ by $x$ by adding the exponents.

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Step 4.3.1.2.1.5.1

Move $x$ .

$1 - x + x - (x \cdot x)$

Step 4.3.1.2.1.5.2

Multiply $x$ by $x$ .

$1 - x + x - x^{2}$

Step 4.3.1.2.2

Add $- x$ and $x$ .

$1 + 0 - x^{2}$

Step 4.3.1.2.3

Add $1$ and $0$ .

$1 - x^{2}$

Step 4.3.1.3

Reorder $1$ and $- x^{2}$ .

$- x^{2} + 1$

Step 4.3.2

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - 1$

$b = 0$

$c = 1$

Step 4.3.3

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 4.3.4

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 4.3.4.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{0}{2 \cdot - 1}$

Step 4.3.4.2

Simplify the right side.

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Step 4.3.4.2.1

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

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Step 4.3.4.2.1.1

Factor $2$ out of $0$ .

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

Step 4.3.4.2.1.2

Move the negative one from the denominator of $\frac{0}{- 1}$ .

$d = - 1 \cdot 0$

Step 4.3.4.2.2

Rewrite $- 1 \cdot 0$ as $- 0$ .

$d = - 0$

Step 4.3.4.2.3

Multiply $- 1$ by $0$ .

$d = 0$

Step 4.3.5

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 4.3.5.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 1 - \frac{0^{2}}{4 \cdot - 1}$

Step 4.3.5.2

Simplify the right side.

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Step 4.3.5.2.1

Simplify each term.

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Step 4.3.5.2.1.1

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

$e = 1 - \frac{0}{4 \cdot - 1}$

Step 4.3.5.2.1.2

Multiply $4$ by $- 1$ .

$e = 1 - \frac{0}{- 4}$

Step 4.3.5.2.1.3

Divide $0$ by $- 4$ .

$e = 1 - 0$

Step 4.3.5.2.1.4

Multiply $- 1$ by $0$ .

$e = 1 + 0$

Step 4.3.5.2.2

Add $1$ and $0$ .

$e = 1$

Step 4.3.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x + 0)}^{2} + 1$ .

$\int_{- 1}^{1} \sqrt{- {(x + 0)}^{2} + 1} d x$

Step 4.4

Let $u_{1} = x + 0$ . Then $d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 4.4.1

Let $u_{1} = x + 0$ . Find $\frac{d u_{1}}{d x}$ .

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Step 4.4.1.1

Differentiate $x + 0$ .

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

Step 4.4.1.2

By the Sum Rule, the derivative of $x + 0$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [0]$ .

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

Step 4.4.1.3

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

$1 + \frac{d}{d x} [0]$

Step 4.4.1.4

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

$1 + 0$

Step 4.4.1.5

Add $1$ and $0$ .

$1$

Step 4.4.2

Substitute the lower limit in for $x$ in $u_{1} = x + 0$ .

$u_{lower} = - 1 + 0$

Step 4.4.3

Add $- 1$ and $0$ .

$u_{lower} = - 1$

Step 4.4.4

Substitute the upper limit in for $x$ in $u_{1} = x + 0$ .

$u_{upper} = 1 + 0$

Step 4.4.5

Add $1$ and $0$ .

$u_{upper} = 1$

Step 4.4.6

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

$u_{lower} = - 1$

$u_{upper} = 1$

Step 4.4.7

Rewrite the problem using $u_{1}$ , $d u_{1}$ , and the new limits of integration.

$\int_{- 1}^{1} \sqrt{- {u_{1}}^{2} + 1} d u_{1}$

Step 4.5

Let $u_{1} = \sin (t)$ , where $- \frac{π}{2} \leq t \leq \frac{π}{2}$ . Then $d u_{1} = \cos (t) d t$ . Note that since $- \frac{π}{2} \leq t \leq \frac{π}{2}$ , $\cos (t)$ is positive.

$\int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{- \sin^{2} (t) + 1} \cos (t) d t$

Step 4.6

Simplify terms.

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Step 4.6.1

Simplify $\sqrt{- \sin^{2} (t) + 1}$ .

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Step 4.6.1.1

Reorder $- \sin^{2} (t)$ and $1$ .

$\int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{1 - \sin^{2} (t)} \cos (t) d t$

Step 4.6.1.2

Apply pythagorean identity.

$\int_{- \frac{π}{2}}^{\frac{π}{2}} \sqrt{\cos^{2} (t)} \cos (t) d t$

Step 4.6.1.3

Pull terms out from under the radical, assuming positive real numbers.

$\int_{- \frac{π}{2}}^{\frac{π}{2}} \cos (t) \cos (t) d t$

Step 4.6.2

Simplify.

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Step 4.6.2.1

Raise $\cos (t)$ to the power of $1$ .

$\int_{- \frac{π}{2}}^{\frac{π}{2}} \cos^{1} (t) \cos (t) d t$

Step 4.6.2.2

Raise $\cos (t)$ to the power of $1$ .

$\int_{- \frac{π}{2}}^{\frac{π}{2}} \cos^{1} (t) \cos^{1} (t) d t$

Step 4.6.2.3

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

$\int_{- \frac{π}{2}}^{\frac{π}{2}} {\cos (t)}^{1 + 1} d t$

Step 4.6.2.4

Add $1$ and $1$ .

$\int_{- \frac{π}{2}}^{\frac{π}{2}} \cos^{2} (t) d t$

Step 4.7

Use the half-angle formula to rewrite $\cos^{2} (t)$ as $\frac{1 + \cos (2 t)}{2}$ .

$\int_{- \frac{π}{2}}^{\frac{π}{2}} \frac{1 + \cos (2 t)}{2} d t$

Step 4.8

Since $\frac{1}{2}$ is constant with respect to $t$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int_{- \frac{π}{2}}^{\frac{π}{2}} 1 + \cos (2 t) d t$

Step 4.9

Split the single integral into multiple integrals.

$\frac{1}{2} (\int_{- \frac{π}{2}}^{\frac{π}{2}} d t + \int_{- \frac{π}{2}}^{\frac{π}{2}} \cos (2 t) d t)$

Step 4.10

Apply the constant rule.

$\frac{1}{2} (t]_{- \frac{π}{2}}^{\frac{π}{2}} + \int_{- \frac{π}{2}}^{\frac{π}{2}} \cos (2 t) d t)$

Step 4.11

Let $u_{2} = 2 t$ . Then $d u_{2} = 2 d t$ , so $\frac{1}{2} d u_{2} = d t$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 4.11.1

Let $u_{2} = 2 t$ . Find $\frac{d u_{2}}{d t}$ .

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Step 4.11.1.1

Differentiate $2 t$ .

$\frac{d}{d t} [2 t]$

Step 4.11.1.2

Since $2$ is constant with respect to $t$ , the derivative of $2 t$ with respect to $t$ is $2 \frac{d}{d t} [t]$ .

$2 \frac{d}{d t} [t]$

Step 4.11.1.3

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

$2 \cdot 1$

Step 4.11.1.4

Multiply $2$ by $1$ .

$2$

Step 4.11.2

Substitute the lower limit in for $t$ in $u_{2} = 2 t$ .

$u_{lower} = 2 (- \frac{π}{2})$

Step 4.11.3

Cancel the common factor of $2$ .

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Step 4.11.3.1

Move the leading negative in $- \frac{π}{2}$ into the numerator.

$u_{lower} = 2 \frac{- π}{2}$

Step 4.11.3.2

Cancel the common factor.

$u_{lower} = 2 \frac{- π}{2}$

Step 4.11.3.3

Rewrite the expression.

$u_{lower} = - π$

Step 4.11.4

Substitute the upper limit in for $t$ in $u_{2} = 2 t$ .

$u_{upper} = 2 \frac{π}{2}$

Step 4.11.5

Cancel the common factor of $2$ .

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Step 4.11.5.1

Cancel the common factor.

$u_{upper} = 2 \frac{π}{2}$

Step 4.11.5.2

Rewrite the expression.

$u_{upper} = π$

Step 4.11.6

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

$u_{lower} = - π$

$u_{upper} = π$

Step 4.11.7

Rewrite the problem using $u_{2}$ , $d u_{2}$ , and the new limits of integration.

$\frac{1}{2} (t]_{- \frac{π}{2}}^{\frac{π}{2}} + \int_{- π}^{π} \cos (u_{2}) \frac{1}{2} d u_{2})$

Step 4.12

Combine $\cos (u_{2})$ and $\frac{1}{2}$ .

$\frac{1}{2} (t]_{- \frac{π}{2}}^{\frac{π}{2}} + \int_{- π}^{π} \frac{\cos (u_{2})}{2} d u_{2})$

Step 4.13

Since $\frac{1}{2}$ is constant with respect to $u_{2}$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} (t]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{1}{2} \int_{- π}^{π} \cos (u_{2}) d u_{2})$

Step 4.14

The integral of $\cos (u_{2})$ with respect to $u_{2}$ is $\sin (u_{2})$ .

$\frac{1}{2} (t]_{- \frac{π}{2}}^{\frac{π}{2}} + \frac{1}{2} \sin (u_{2})]_{- π}^{π})$

Step 4.15

Substitute and simplify.

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Step 4.15.1

Evaluate $t$ at $\frac{π}{2}$ and at $- \frac{π}{2}$ .

$\frac{1}{2} ((\frac{π}{2}) + \frac{π}{2} + \frac{1}{2} \sin (u_{2})]_{- π}^{π})$

Step 4.15.2

Evaluate $\sin (u_{2})$ at $π$ and at $- π$ .

$\frac{1}{2} ((\frac{π}{2}) + \frac{π}{2} + \frac{1}{2} ((\sin (π)) - \sin (- π)))$

Step 4.15.3

Simplify.

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Step 4.15.3.1

Combine the numerators over the common denominator.

$\frac{1}{2} (\frac{π + π}{2} + \frac{1}{2} ((\sin (π)) - \sin (- π)))$

Step 4.15.3.2

Add $π$ and $π$ .

$\frac{1}{2} (\frac{2 π}{2} + \frac{1}{2} ((\sin (π)) - \sin (- π)))$

Step 4.15.3.3

Cancel the common factor of $2$ .

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Step 4.15.3.3.1

Cancel the common factor.

$\frac{1}{2} (\frac{2 π}{2} + \frac{1}{2} ((\sin (π)) - \sin (- π)))$

Step 4.15.3.3.2

Divide $π$ by $1$ .

$\frac{1}{2} (π + \frac{1}{2} ((\sin (π)) - \sin (- π)))$

$\frac{1}{2} (π + \frac{1}{2} (\sin (π) - \sin (- π)))$

Step 4.16

Simplify.

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Step 4.16.1

Simplify each term.

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Step 4.16.1.1

Simplify each term.

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Step 4.16.1.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

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

Step 4.16.1.1.2

The exact value of $\sin (0)$ is $0$ .

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

Step 4.16.1.1.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

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

Step 4.16.1.1.4

The exact value of $\sin (0)$ is $0$ .

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

Step 4.16.1.1.5

Multiply $- 1$ by $0$ .

$\frac{1}{2} (π + \frac{1}{2} (0 + 0))$

Step 4.16.1.2

Add $0$ and $0$ .

$\frac{1}{2} (π + \frac{1}{2} \cdot 0)$

Step 4.16.1.3

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

$\frac{1}{2} (π + 0)$

Step 4.16.2

Add $π$ and $0$ .

$\frac{1}{2} π$

Step 4.16.3

Combine $\frac{1}{2}$ and $π$ .

$\frac{π}{2}$

Step 5