Can the value of pi be found through integration?

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In summary, the conversation discusses the possibility of finding an exact solution for the integral ∫√(1-x*x)dx from 0 to 1 and the use of a trig substitution to evaluate it. The answer is found to be 1/4pi or &pi/4, depending on the method used. The concept of a trig substitution is briefly mentioned and recommended to be explored further.
  • #1
AndersHermansson
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Is it possible to find an exact solution for this integral?

∫√(1-x*x)dx from 0 to 1

Is it possible to differentiate a root expression?

I found that:
pi = 4 * ∫√(1-x*x)dx from 0 to 1
 
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  • #2
the equation is for a semicircle of radius 1 from 0 to 1 you get a quarter circle and 1/4 pi *r^2=1/4pi

1/4pi is the answer
you could also evaluate the integral using a trig subsitution

you already found the answer if you divide both sides of the equation by 4 in your solution you also get the answer
 
Last edited:
  • #3
for the trig sub

x=sin(θ)
dx=cos(θ)dθ
substitute into original integral simpligy trig expresion and switch limits of integration (evaluate interms of theta) and you will get &pi/4

if you haven't learned trig subs check it out in your calc book it not a very hard topic.
 
  • #4
what is x*? is x a complex variable? i don t quite understand your integral
 
  • #5
Probably because it's a lot easier than you're used to. :)
 

1. What is an integral?

An integral is a mathematical concept that represents the area under a curve on a graph. It is also known as the antiderivative of a function.

2. Why is it important to find a solution for an integral?

Solving an integral allows us to find the exact area under a curve, which is often necessary in many fields such as physics, engineering, and economics.

3. How do you solve an integral?

To solve an integral, you can use various techniques such as substitution, integration by parts, or trigonometric identities. It also depends on the type of integral, whether it is definite or indefinite.

4. What are the applications of integrals in real life?

Integrals have many applications in real life, such as calculating volumes and areas of irregular shapes, finding the average value of a function, and determining the displacement of an object over time.

5. Are there any tips for solving integrals?

Some tips for solving integrals include identifying the type of integral, using appropriate techniques, and practicing regularly to improve your skills. It is also helpful to break down the integral into smaller parts and use algebraic manipulation to simplify it.

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