# What is the mistake in calculating the integral of the absolute sine function?

• dirk_mec1
In summary, the conversation was about finding the area under the absolute value of a sine function from 0 to 2018π. The solution involved determining the period of the function and counting the number of "humps" to find the total area. The final answer was off by a factor of 2018, but the mistake was later corrected. The conversation also briefly touched on the integral of sin(2018x) and the area of each "hump" of the sine function.
dirk_mec1

## Homework Statement

$$\int_0^{2018 \pi} \lvert \sin(2018x) \lvert \mbox{d}x$$

## The Attempt at a Solution

So the period is:
$$\frac{2 \pi}{ 2018}$$

Each "hump" of the sine has an area of 2 so if I count the number of humps I am done. In one period of an absolute sine function the area is thus 4.

So the requested area is:

$$4 \cdot \frac{2018 \pi}{\frac{2 \pi}{ 2018} } = 2 \cdot 2018^2$$

I am off by a factor of 2018. Where is my mistake?

dirk_mec1 said:

## Homework Statement

$$\int_0^{2018 \pi} \lvert \sin(2018x) \lvert \mbox{d}x$$

## The Attempt at a Solution

So the period is:
$$\frac{2 \pi}{ 2018}$$

Each "hump" of the sine has an area of 2 so if I count the number of humps I am done. In one period of an absolute sine function the area is thus 4.

So the requested area is:

$$4 \cdot \frac{2018 \pi}{\frac{2 \pi}{ 2018} } = 2 \cdot 2018^2$$

I am off by a factor of 2018. Where is my mistake?
What is ##\int sin(2018x) dx##?

1/2018 * -cos(2018x) + C.

dirk_mec1 said:
Each "hump" of the sine has an area of 2
Can you prove this?

Thanks. I understand my mistake!

## 1. What is the integral of the absolute sine function?

The integral of the absolute sine function is the area under the curve of the absolute value of the sine function. It is denoted by ∫|sin(x)| dx and can be evaluated using various integration techniques.

## 2. How is the integral of the absolute sine function different from the regular sine function?

The main difference between the two is that the absolute sine function takes the absolute value of the sine function, which means all negative values are turned into positive values. This results in a graph that is symmetrical about the x-axis.

## 3. What is the importance of the integral of the absolute sine function?

The integral of the absolute sine function is important in various fields of science and engineering, including physics, signal processing, and electrical engineering. It is used to calculate the displacement, velocity, and acceleration of objects in motion.

## 4. How is the integral of the absolute sine function calculated?

The integral of the absolute sine function can be calculated using integration techniques such as substitution, integration by parts, or trigonometric identities. The specific method used will depend on the complexity of the function and the desired level of accuracy.

## 5. Are there any real-life applications of the integral of the absolute sine function?

Yes, the integral of the absolute sine function has many real-life applications. For example, it is used in calculating the area under a sound wave to determine the loudness of a sound. It is also used in calculating the displacement of a vibrating object in engineering and physics problems.

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