Calculus Examples

$y = \sin (x)$ , $y = 5 x$ , $x = \frac{π}{2}$ , $x = π$

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.

$\sin (x) = 5 x$

Step 1.2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x = 0$

Step 1.3

Evaluate $y$ when $x = 0$ .

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Step 1.3.1

Substitute $0$ for $x$ .

$y = 5 (0)$

Step 1.3.2

Substitute $0$ for $x$ in $y = 5 (0)$ and solve for $y$ .

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Step 1.3.2.1

Remove parentheses.

$y = 5 (0)$

Step 1.3.2.2

Multiply $5$ by $0$ .

$y = 0$

Step 1.4

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

$(0, 0)$

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.

$A r e a = \int_{\frac{π}{2}}^{π} 5 x d x - \int_{\frac{π}{2}}^{π} \sin (x) d x$

Step 3

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

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Step 3.1

Combine the integrals into a single integral.

$\int_{\frac{π}{2}}^{π} 5 x - (\sin (x)) d x$

Step 3.2

Multiply $- 1$ by $\sin (x)$ .

$\int_{\frac{π}{2}}^{π} 5 x - \sin (x) d x$

Step 3.3

Split the single integral into multiple integrals.

$\int_{\frac{π}{2}}^{π} 5 x d x + \int_{\frac{π}{2}}^{π} - \sin (x) d x$

Step 3.4

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$5 \int_{\frac{π}{2}}^{π} x d x + \int_{\frac{π}{2}}^{π} - \sin (x) d x$

Step 3.5

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

$5 (\frac{1}{2} x^{2}]_{\frac{π}{2}}^{π}) + \int_{\frac{π}{2}}^{π} - \sin (x) d x$

Step 3.6

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

$5 (\frac{x^{2}}{2}]_{\frac{π}{2}}^{π}) + \int_{\frac{π}{2}}^{π} - \sin (x) d x$

Step 3.7

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

$5 (\frac{x^{2}}{2}]_{\frac{π}{2}}^{π}) - \int_{\frac{π}{2}}^{π} \sin (x) d x$

Step 3.8

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

$5 (\frac{x^{2}}{2}]_{\frac{π}{2}}^{π}) - (- \cos (x)]_{\frac{π}{2}}^{π})$

Step 3.9

Simplify the answer.

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Step 3.9.1

Substitute and simplify.

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Step 3.9.1.1

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

$5 ((\frac{π^{2}}{2}) - \frac{{(\frac{π}{2})}^{2}}{2}) - (- \cos (x)]_{\frac{π}{2}}^{π})$

Step 3.9.1.2

Evaluate $- \cos (x)$ at $π$ and at $\frac{π}{2}$ .

$5 (\frac{π^{2}}{2} - \frac{{(\frac{π}{2})}^{2}}{2}) - (- \cos (π) + \cos (\frac{π}{2}))$

Step 3.9.2

Simplify.

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Step 3.9.2.1

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

$5 (\frac{π^{2}}{2} - \frac{{(\frac{π}{2})}^{2}}{2}) - (- \cos (π) + 0)$

Step 3.9.2.2

Add $- \cos (π)$ and $0$ .

$5 (\frac{π^{2}}{2} - \frac{{(\frac{π}{2})}^{2}}{2}) - - \cos (π)$

Step 3.9.2.3

Multiply $- 1$ by $- 1$ .

$5 (\frac{π^{2}}{2} - \frac{{(\frac{π}{2})}^{2}}{2}) + 1 \cos (π)$

Step 3.9.2.4

Multiply $\cos (π)$ by $1$ .

$5 (\frac{π^{2}}{2} - \frac{{(\frac{π}{2})}^{2}}{2}) + \cos (π)$

Step 3.9.3

Simplify.

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Step 3.9.3.1

Simplify the numerator.

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Step 3.9.3.1.1

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

$5 (\frac{π^{2}}{2} - \frac{\frac{π^{2}}{2^{2}}}{2}) + \cos (π)$

Step 3.9.3.1.2

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

$5 (\frac{π^{2}}{2} - \frac{\frac{π^{2}}{4}}{2}) + \cos (π)$

Step 3.9.3.2

Multiply the numerator by the reciprocal of the denominator.

$5 (\frac{π^{2}}{2} - (\frac{π^{2}}{4} \cdot \frac{1}{2})) + \cos (π)$

Step 3.9.3.3

Combine.

$5 (\frac{π^{2}}{2} - \frac{π^{2} \cdot 1}{4 \cdot 2}) + \cos (π)$

Step 3.9.3.4

Multiply $π^{2}$ by $1$ .

$5 (\frac{π^{2}}{2} - \frac{π^{2}}{4 \cdot 2}) + \cos (π)$

Step 3.9.3.5

Multiply $4$ by $2$ .

$5 (\frac{π^{2}}{2} - \frac{π^{2}}{8}) + \cos (π)$

Step 3.9.3.6

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

$5 (\frac{π^{2}}{2} \cdot \frac{4}{4} - \frac{π^{2}}{8}) + \cos (π)$

Step 3.9.3.7

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

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Step 3.9.3.7.1

Multiply $\frac{π^{2}}{2}$ by $\frac{4}{4}$ .

$5 (\frac{π^{2} \cdot 4}{2 \cdot 4} - \frac{π^{2}}{8}) + \cos (π)$

Step 3.9.3.7.2

Multiply $2$ by $4$ .

$5 (\frac{π^{2} \cdot 4}{8} - \frac{π^{2}}{8}) + \cos (π)$

Step 3.9.3.8

Combine the numerators over the common denominator.

$5 \frac{π^{2} \cdot 4 - π^{2}}{8} + \cos (π)$

Step 3.9.3.9

Move $4$ to the left of $π^{2}$ .

$5 \frac{4 \cdot π^{2} - π^{2}}{8} + \cos (π)$

Step 3.9.3.10

Subtract $π^{2}$ from $4 π^{2}$ .

$5 \frac{3 π^{2}}{8} + \cos (π)$

Step 3.9.3.11

Multiply $5 \frac{3 π^{2}}{8}$ .

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Step 3.9.3.11.1

Combine $5$ and $\frac{3 π^{2}}{8}$ .

$\frac{5 (3 π^{2})}{8} + \cos (π)$

Step 3.9.3.11.2

Multiply $3$ by $5$ .

$\frac{15 π^{2}}{8} + \cos (π)$

Step 3.10

Simplify each term.

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Step 3.10.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the second quadrant.

$\frac{15 π^{2}}{8} - \cos (0)$

Step 3.10.2

The exact value of $\cos (0)$ is $1$ .

$\frac{15 π^{2}}{8} - 1 \cdot 1$

Step 3.10.3

Multiply $- 1$ by $1$ .

$\frac{15 π^{2}}{8} - 1$

Step 4