Circumference of a circle in parametrics

In summary, the conversation discusses the process of finding the length of a curve using the formula for arc length. The formula involves calculating the square root of the sum of the squares of the derivatives with respect to either x or t. The discussion also mentions the issue of undefined values and the need for a trig substitution to solve the integral. The conversation concludes by mentioning the teacher's requirement to solve the problem using both parametrization and traditional methods.
  • #1
hangainlover
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Homework Statement



x^2+y^2=1

Homework Equations



length of curve square root (1+(dx/dy)^2)dy or square root ((dx/dt)^2 + (dy/dt)^2) dt




3. The Attempt at a Solution [/b

I isolated y and got y= square root (1-x^2)

finding dy/dx = -x/square root (1-x^2)
i plugged that into the formula but i did not get the correct answer.

im aware that if i do integral of square root (1+ (dy/dx)^2 ) i have to do imporper integral as i get undefined for denominator at x=-1,1 so what should i do
we all know that answer should be 2pi
 
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  • #2
Ignore the 'undefined' issue. Just do a trig substitution to solve the integral. Once you've done the trig substitution you won't see the singularity. BTW you'll get pi, not 2*pi. Your parametrization only covers half the circle.
 
  • #3
I know, i can get the answer by defining the circle in trig.

But, shouldn't i be able to do it in conventional way and by paramatrization as well?
i just need to take care of the x values and t values where the function gets undefined.
My teacher wants me to do it in these two ways.
 
  • #4
Even to do it in the x-y way you need to integrate 1/sqrt(1-x^2) from -1 to 1, right? Even if you haven't parametrized it as a trig function you still need to do a trig substitution to solve that integral. That's what I'm talking about.
 

What is the formula for finding the circumference of a circle in parametrics?

The formula for finding the circumference of a circle in parametrics is C = 2πr, where r is the radius of the circle.

How is parametrics used in calculating the circumference of a circle?

Parametrics is used to describe the motion of a point on the circumference of a circle in terms of its position, velocity, and acceleration. This information can then be used to calculate the circumference using the formula C = 2πr.

Can parametrics be used to find the circumference of a circle with a given center and radius?

Yes, parametrics can be used to find the circumference of a circle with a given center and radius. The formula C = 2πr can be used to calculate the circumference by substituting the given radius for r.

What is the relationship between the circumference of a circle and its diameter in parametrics?

In parametrics, the circumference of a circle is directly proportional to its diameter. This means that the ratio of the circumference to the diameter is always equal to π, or approximately 3.14.

How does parametrics help in understanding the concept of circles and their circumferences?

Parametrics helps in understanding the concept of circles and their circumferences by providing a mathematical model for describing the motion of points on the circumference of a circle. This allows us to calculate the circumference and understand how it is related to other properties of a circle, such as its diameter and radius.

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