# Area under a curve-positive and negative area cancellation

• Krushnaraj Pandya
In summary, the problem is finding the area bounded by sin(x) and cos(x) between pi/4 and 5pi/4, and the solution involves determining the intervals where sin(x) - cos(x) is positive or negative and taking the absolute value of the portion that lies below the horizontal axis. This concept also applies to simpler examples, such as finding the area under the graph of sin(x) between 0 and 2pi.
Krushnaraj Pandya
Gold Member

## Homework Statement

So this is a problem I've been facing while finding the area under some curves, for example finding the area bounded by sinx and cosx between pi/4 and 5pi/4. integrating sinx-cosx with these limits would result in the positive and negative areas cancelling out, how do I get the modulus of this area. In basic functions like sinx I would just double the area I found between 0 and pi to calculate upto 2pi but it seems more troubling to do here.
Do I need to integrate the modulus of the function, then I'd have to do extra work and find where sinx-cosx changes sign

## Homework Equations

All pertaining to calculus

## The Attempt at a Solution

Given above within the problem

Krushnaraj Pandya said:

## Homework Statement

So this is a problem I've been facing while finding the area under some curves, for example finding the area bounded by sinx and cosx between pi/4 and 5pi/4.
What does "bounded by sinx and cosx" mean?
Is this your integral? ##\int_{\frac \pi 4}^{\frac{5\pi} 4} \sin(x) - \cos(x) dx##

Krushnaraj Pandya said:
integrating sinx-cosx with these limits would result in the positive and negative areas cancelling out, how do I get the modulus of this area. In basic functions like sinx I would just double the area I found between 0 and pi to calculate upto 2pi but it seems more troubling to do here.
Do I need to integrate the modulus of the function, then I'd have to do extra work and find where sinx-cosx changes sign
You need to determine the intervals on which ##\sin(x) - \cos(x)## is positive, and those on which this expression is negative.

As a simpler example, ##\int_0^{2\pi} \sin(x) dx = 0##, but if you want to find the area bounded by the graph of this function, you need the absolute value of the portion that lies below the horizontal axis. It's the same idea with the function you want to integrate.

Krushnaraj Pandya said:

## Homework Equations

All pertaining to calculus

## The Attempt at a Solution

Given above within the problem

Mark44 said:
What does "bounded by sinx and cosx" mean?
Is this your integral? ∫5π4π4sin(x)−cos(x)dx∫π45π4sin⁡(x)−cos⁡(x)dx\int_{\frac \pi 4}^{\frac{5\pi} 4} \sin(x) - \cos(x) dx
yes it is, I got the correct answer as well.
Mark44 said:
You need to determine the intervals on which sin(x)−cos(x)sin⁡(x)−cos⁡(x)\sin(x) - \cos(x) is positive, and those on which this expression is negative.

As a simpler example, ∫2π0sin(x)dx=0∫02πsin⁡(x)dx=0\int_0^{2\pi} \sin(x) dx = 0, but if you want to find the area bounded by the graph of this function, you need the absolute value of the portion that lies below the horizontal axis. It's the same idea with the function you want to integrate.
Got it, thank you very much :D

## 1. What does "area under a curve" mean?

The area under a curve refers to the total area that is enclosed by a curve on a graph and the x-axis. This area can be positive or negative depending on the position of the curve in relation to the x-axis.

## 2. How is the area under a curve calculated?

The area under a curve can be calculated using integration techniques in calculus. It involves breaking down the curve into small segments and finding the total area of each segment, then adding them together to get the total area under the curve.

## 3. What is meant by "positive and negative area cancellation"?

Positive and negative area cancellation refers to the phenomenon where positive and negative areas under a curve cancel each other out, resulting in a net area of zero. This can occur when the curve crosses the x-axis multiple times, resulting in both positive and negative areas.

## 4. Why is the concept of area under a curve important?

The concept of area under a curve is important in many fields of science, such as physics, engineering, and economics. It allows us to calculate important quantities like displacement, work, and profit, which are essential for understanding and solving real-world problems.

## 5. How can we use the area under a curve to make predictions or analyze data?

The area under a curve can be used to make predictions or analyze data by looking at the overall shape of the curve and the distribution of positive and negative areas. This can give us insights into patterns, trends, and relationships between variables, helping us make informed decisions and predictions.

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