Integral inequality with sin proof

In summary, the integral inequality with sin proof is a mathematical inequality used in calculus and analysis to relate the integral of a function to the sine of that function. It is derived using mathematical induction and has various real-life applications in physics, engineering, and other scientific fields. However, it can only be applied to certain types of functions that meet specific conditions. Additionally, there are variations of this inequality, such as the Cauchy-Schwarz and Hölder's inequalities, which may be more suitable for different types of functions or problems.
  • #1
ptolema
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Homework Statement



prove that
integral.jpg
for all x>0

Homework Equations



-1 [tex]\leq[/tex] sin t [tex]\leq[/tex] 1

The Attempt at a Solution


the area under the graph is increasing as x increases
also, i tried to write it the sigma way:
leibsigma-1.jpg
then take the limit as n-->infinity
i got stuck trying to figure out how to work with sine in sigma notation, but I'm not even sure if my attempt would get anywhere

can anyone give me any pointers on how to do this?
 
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  • #2
As x increases, when will the value of the integral be increasing, when will it be decreasing?
 

1. What is the integral inequality with sin proof?

The integral inequality with sin proof is a mathematical inequality that relates the integral of a function to the sine of that function. It is commonly used in calculus and analysis to solve problems involving integrals.

2. How is the integral inequality with sin proof derived?

The integral inequality with sin proof is typically derived using mathematical induction and the properties of integrals and trigonometric functions. It is a well-known result in mathematics and has many applications in various fields.

3. Can the integral inequality with sin proof be applied to any function?

No, the integral inequality with sin proof can only be applied to functions that are continuous and differentiable on a closed interval. Additionally, the function must also satisfy certain conditions, such as being bounded and having a finite integral on the interval.

4. What are some real-life applications of the integral inequality with sin proof?

The integral inequality with sin proof has many applications in physics, engineering, and other scientific fields. It is commonly used to solve problems involving motion, oscillations, and vibrations, as well as in the analysis of electrical circuits and signal processing.

5. Are there any variations of the integral inequality with sin proof?

Yes, there are various versions of the integral inequality with sin proof, including the Cauchy-Schwarz and Hölder's inequalities. These variations have different forms and may be more suitable for certain types of functions or problems.

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