Inequality involving area under a curve

In summary, "inequality involving area under a curve" refers to comparing the sizes of areas formed under a curve on a graph. The area under a curve is typically calculated using integral calculus or numerical methods. Studying this concept allows us to better understand and analyze various aspects of society and can be applied to fields such as economics and sociology. It can also help address social and economic issues by identifying areas of inequality and informing policies and strategies for more equitable distribution of resources.
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anemone
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Prove that for every $x\in (0,\,1)$ the following inequality holds:

$\displaystyle \int_0^1 \sqrt{1+(\cos y)^2} dy>\sqrt{x^2+(\sin x)^2}$
 
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Clearly $\displaystyle \int_0^1 \sqrt{1+(\cos y)^2} dy \ge \int_0^x \sqrt{1+(\cos y)^2} dy$ for each fixed $x\in (0,\,1)$. Observe that $\displaystyle \int_0^x \sqrt{1+(\cos y)^2} dy$ is the arc length of the function $f(y)=\sin y$ on the interval $[0,\,x]$ which is clearly strictly greater than the length of the straight line between the points $(0,\,0)$ and $(x,\, \sin x)$ which in turn is equal to $\sqrt{x^2+(\sin x)^2}$.
 

FAQ: Inequality involving area under a curve

1. What is "inequality involving area under a curve"?

"Inequality involving area under a curve" refers to a mathematical concept that involves finding the area under a curve on a graph and using it to determine the relationship between two variables. It is often used in calculus and other branches of mathematics to analyze and compare different sets of data.

2. How is the area under a curve related to inequality?

The area under a curve can be used to determine the magnitude of the inequality between two variables. If the area under one curve is larger than the area under another curve, it indicates a greater difference between the two variables, and thus a greater level of inequality.

3. What are some real-life applications of "inequality involving area under a curve"?

This concept is commonly used in economics, where it can be used to analyze income distribution and wealth inequality. It is also used in physics and engineering to analyze the relationship between variables such as force and displacement.

4. How is "inequality involving area under a curve" calculated?

To calculate the area under a curve, you can use integration techniques from calculus. This involves breaking the curve into small, manageable sections and using a formula to find the area of each section. The sum of these areas will give you the total area under the curve.

5. What are some limitations of using "inequality involving area under a curve" to measure inequality?

While this concept can provide valuable insights into the relationship between variables, it may not always accurately reflect the full extent of inequality. It is also important to consider other factors and variables that may influence the data and to use multiple methods of analysis for a more comprehensive understanding of inequality.

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