Calculus Examples

$y = \cos (11 x)$ , $y = 0$ , $x = \frac{π}{22}$ , $x = \frac{π}{11}$

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.

$\cos (11 x) = 0$

Step 1.2

Solve $\cos (11 x) = 0$ for $x$ .

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Step 1.2.1

Take the inverse cosine of both sides of the equation to extract $x$ from inside the cosine.

$11 x = \arccos (0)$

Step 1.2.2

Simplify the right side.

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Step 1.2.2.1

The exact value of $\arccos (0)$ is $\frac{π}{2}$ .

$11 x = \frac{π}{2}$

Step 1.2.3

Divide each term in $11 x = \frac{π}{2}$ by $11$ and simplify.

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Step 1.2.3.1

Divide each term in $11 x = \frac{π}{2}$ by $11$ .

$\frac{11 x}{11} = \frac{\frac{π}{2}}{11}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $11$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{11 x}{11} = \frac{\frac{π}{2}}{11}$

Step 1.2.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{π}{2}}{11}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{π}{2} \cdot \frac{1}{11}$

Step 1.2.3.3.2

Multiply $\frac{π}{2} \cdot \frac{1}{11}$ .

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Step 1.2.3.3.2.1

Multiply $\frac{π}{2}$ by $\frac{1}{11}$ .

$x = \frac{π}{2 \cdot 11}$

Step 1.2.3.3.2.2

Multiply $2$ by $11$ .

$x = \frac{π}{22}$

Step 1.2.4

The cosine function is positive in the first and fourth quadrants. To find the second solution, subtract the reference angle from $2 π$ to find the solution in the fourth quadrant.

$11 x = 2 π - \frac{π}{2}$

Step 1.2.5

Solve for $x$ .

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Step 1.2.5.1

Simplify.

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Step 1.2.5.1.1

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

$11 x = 2 π \cdot \frac{2}{2} - \frac{π}{2}$

Step 1.2.5.1.2

Combine $2 π$ and $\frac{2}{2}$ .

$11 x = \frac{2 π \cdot 2}{2} - \frac{π}{2}$

Step 1.2.5.1.3

Combine the numerators over the common denominator.

$11 x = \frac{2 π \cdot 2 - π}{2}$

Step 1.2.5.1.4

Multiply $2$ by $2$ .

$11 x = \frac{4 π - π}{2}$

Step 1.2.5.1.5

Subtract $π$ from $4 π$ .

$11 x = \frac{3 π}{2}$

Step 1.2.5.2

Divide each term in $11 x = \frac{3 π}{2}$ by $11$ and simplify.

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Step 1.2.5.2.1

Divide each term in $11 x = \frac{3 π}{2}$ by $11$ .

$\frac{11 x}{11} = \frac{\frac{3 π}{2}}{11}$

Step 1.2.5.2.2

Simplify the left side.

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Step 1.2.5.2.2.1

Cancel the common factor of $11$ .

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Step 1.2.5.2.2.1.1

Cancel the common factor.

$\frac{11 x}{11} = \frac{\frac{3 π}{2}}{11}$

Step 1.2.5.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{\frac{3 π}{2}}{11}$

Step 1.2.5.2.3

Simplify the right side.

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Step 1.2.5.2.3.1

Multiply the numerator by the reciprocal of the denominator.

$x = \frac{3 π}{2} \cdot \frac{1}{11}$

Step 1.2.5.2.3.2

Multiply $\frac{3 π}{2} \cdot \frac{1}{11}$ .

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Step 1.2.5.2.3.2.1

Multiply $\frac{3 π}{2}$ by $\frac{1}{11}$ .

$x = \frac{3 π}{2 \cdot 11}$

Step 1.2.5.2.3.2.2

Multiply $2$ by $11$ .

$x = \frac{3 π}{22}$

Step 1.2.6

Find the period of $\cos (11 x)$ .

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Step 1.2.6.1

The period of the function can be calculated using $\frac{2 π}{| b |}$ .

$\frac{2 π}{| b |}$

Step 1.2.6.2

Replace $b$ with $11$ in the formula for period.

$\frac{2 π}{| 11 |}$

Step 1.2.6.3

The absolute value is the distance between a number and zero. The distance between $0$ and $11$ is $11$ .

$\frac{2 π}{11}$

Step 1.2.7

The period of the $\cos (11 x)$ function is $\frac{2 π}{11}$ so values will repeat every $\frac{2 π}{11}$ radians in both directions.

$x = \frac{π}{22} + \frac{2 π n}{11}, \frac{3 π}{22} + \frac{2 π n}{11}$ , for any integer $n$

Step 1.2.8

Consolidate the answers.

$x = \frac{π}{22} + \frac{π n}{11}$ , for any integer $n$

Step 1.3

Substitute $\frac{π}{22} + \frac{π n}{11}$ for $x$ .

$y = 0$

Step 1.4

List all of the solutions.

$y = 0, x = \frac{π}{22} + \frac{π n}{11}$

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.

$A r e a = \int_{\frac{π}{22}}^{\frac{π}{11}} 0 d x - \int_{\frac{π}{22}}^{\frac{π}{11}} \cos (11 x) d x$

Step 3

Integrate to find the area between $\frac{π}{22}$ and $\frac{π}{11}$ .

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Step 3.1

Combine the integrals into a single integral.

$\int_{\frac{π}{22}}^{\frac{π}{11}} 0 - (\cos (11 x)) d x$

Step 3.2

Subtract $\cos (11 x)$ from $0$ .

$\int_{\frac{π}{22}}^{\frac{π}{11}} - \cos (11 x) d x$

Step 3.3

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

$- \int_{\frac{π}{22}}^{\frac{π}{11}} \cos (11 x) d x$

Step 3.4

Let $u = 11 x$ . Then $d u = 11 d x$ , so $\frac{1}{11} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.4.1

Let $u = 11 x$ . Find $\frac{d u}{d x}$ .

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Step 3.4.1.1

Differentiate $11 x$ .

$\frac{d}{d x} [11 x]$

Step 3.4.1.2

Since $11$ is constant with respect to $x$ , the derivative of $11 x$ with respect to $x$ is $11 \frac{d}{d x} [x]$ .

$11 \frac{d}{d x} [x]$

Step 3.4.1.3

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

$11 \cdot 1$

Step 3.4.1.4

Multiply $11$ by $1$ .

$11$

Step 3.4.2

Substitute the lower limit in for $x$ in $u = 11 x$ .

$u_{lower} = 11 \frac{π}{22}$

Step 3.4.3

Cancel the common factor of $11$ .

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Step 3.4.3.1

Factor $11$ out of $22$ .

$u_{lower} = 11 \frac{π}{11 (2)}$

Step 3.4.3.2

Cancel the common factor.

$u_{lower} = 11 \frac{π}{11 \cdot 2}$

Step 3.4.3.3

Rewrite the expression.

$u_{lower} = \frac{π}{2}$

Step 3.4.4

Substitute the upper limit in for $x$ in $u = 11 x$ .

$u_{upper} = 11 \frac{π}{11}$

Step 3.4.5

Cancel the common factor of $11$ .

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Step 3.4.5.1

Cancel the common factor.

$u_{upper} = 11 \frac{π}{11}$

Step 3.4.5.2

Rewrite the expression.

$u_{upper} = π$

Step 3.4.6

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

$u_{lower} = \frac{π}{2}$

$u_{upper} = π$

Step 3.4.7

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

$- \int_{\frac{π}{2}}^{π} \cos (u) \frac{1}{11} d u$

Step 3.5

Combine $\cos (u)$ and $\frac{1}{11}$ .

$- \int_{\frac{π}{2}}^{π} \frac{\cos (u)}{11} d u$

Step 3.6

Since $\frac{1}{11}$ is constant with respect to $u$ , move $\frac{1}{11}$ out of the integral.

$- (\frac{1}{11} \int_{\frac{π}{2}}^{π} \cos (u) d u)$

Step 3.7

The integral of $\cos (u)$ with respect to $u$ is $\sin (u)$ .

$- \frac{1}{11} \sin (u)]_{\frac{π}{2}}^{π}$

Step 3.8

Evaluate $\sin (u)$ at $π$ and at $\frac{π}{2}$ .

$- \frac{1}{11} (\sin (π) - \sin (\frac{π}{2}))$

Step 3.9

Simplify.

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Step 3.9.1

The exact value of $\sin (\frac{π}{2})$ is $1$ .

$- \frac{1}{11} (\sin (π) - 1 \cdot 1)$

Step 3.9.2

Multiply $- 1$ by $1$ .

$- \frac{1}{11} (\sin (π) - 1)$

Step 3.10

Simplify.

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Step 3.10.1

Simplify each term.

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Step 3.10.1.1

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

$- \frac{1}{11} (\sin (0) - 1)$

Step 3.10.1.2

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

$- \frac{1}{11} (0 - 1)$

Step 3.10.2

Subtract $1$ from $0$ .

$- \frac{1}{11} \cdot - 1$

Step 3.10.3

Multiply $- \frac{1}{11} \cdot - 1$ .

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Step 3.10.3.1

Multiply $- 1$ by $- 1$ .

$1 (\frac{1}{11})$

Step 3.10.3.2

Multiply $\frac{1}{11}$ by $1$ .

$\frac{1}{11}$

Step 4