Calculating the perimeter of a region using integrals

In summary, the length of a perimeter (also known as the arclength) of a region in the plane can be calculated using integral calculus. The formula for this is ds = sqrt(1 + (dy/dx)^2)dx, where dx and dy represent small changes in x and y, respectively. This formula is derived from Pythagoras' theorem and is used to approximate the perimeter by hugging the curve with small straight lines. To get a more accurate result, the integral is used to sum up all the tiny lengths. The formula for calculating the perimeter of a function y(x) from x=a to x=b is given by \int^b_a \sqrt{1+ (y')^2} dx.
  • #1
esmeco
144
0
I was wondering,how do we calculate the perimeter of a region using integral calculus?I know that to calculate the area we have to draw the region and if we want the reunion of those regions we have to sum them and use some values to define the boundaries of the integral.
Any help on this is really appreciated!Thanks in advance!
 
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  • #2
In the most general terms, the length of a perimeter (from the way you stated the question, I am assuming you are talking about a region of the plane, where the perimeter is a curve) is given as the integral of ds,where:

ds2=dx2+dy2.

In practice, you would use:

ds=(1+(dy/dx)2)1/2dx
 
  • #3
In other words, you are talking about arclength. Use mathman's formula or, if x and y are given in terms of a parameter, t, use
[tex]ds= \sqrt{\left(\frac{dx}{dt}\right)+ \left(\frac{dy}{dt}\right)^2}ds[/tex]

If the perimeter is given by two or more different formula (as a region defined to be between two curves) you will need to integrate each separately and then add.
 
  • #4
Sorry for my "ignorence" but, what does de ds means?Is it the integral?
 
  • #5
No, "ds" is the length of the tiny tangent vector at one point of the curve.
We add all those tiny lengths up to get the entire curve's length.
 
  • #6
I have a feeling esmeco wants a appraoch similar to the way he posted the integral for areas.

For the area, we cut up the region into tiny rectangles yes? well the integral basically makes it perfectly accurate by making lots and lots of rectanles. For that to happen, the base has to be smaller. When its really small, we call it dx. a small change in x. a small change in height would then be dy.

if we took those small changes in x and y, and applied pythagoras to them to get the straight line that connects them. That imitates the perimeter abit. Then we integrate it to sum all of the small ones. :)
 
  • #7
So, if I have a function like 0<=y<=sqrt(x+2) to find the perimeter of that region I have to make a tangent to the curve and use the pithagoras formula?Like in the case,the length of the catects would be two, so it should be something like:

ds=sqrt(2^2 + 2^2)=sqrt8

Is this correct?But what do I do now with the hipothenus value?
 
  • #8
Im going to ignore the < signs and the 0 because its not longer a function then.

if [tex]y=\sqrt{x+2}[/tex]. Try drawing the graph. Then zoom in. It gets abit straighter yea? Then in even more, it'll look even more straight. Since a tangent straight line then becomes a good approximation of that part, we make a straight line that hugs the curve. To do that, we take the lines vertical height, and horizontal height. From those 2, and Pythagoras Theorem, we get the length of the line that hugs the curve. The tangent will vary in different places, and this method of doing this is very long, taxing and sort of in accurate. With Integral calculus, we can do this process perfectly accurately and very quickly.

But you haven't done Differential Calculus yet, so your at least a good few months off this.
 
  • #9
Um actually I've seen some of your other posts, and it seems you might be further ahead in mathematics than i am lol. If your doing differential equations and asking for the primitive of sin^3 x, perhaps you'll understand this formula.

[tex]\int^b_a \sqrt{1+ (y')^2} dx[/tex]. Thats the perimeter of the function y, from b to a. I tried use prime notation rather than Leibniz, but that's hard to do with Integrals lol.

From your other forums, it seems you haven't encountered integrals, but rather only primitives.
 
  • #10
That's the length of the graph of y=y(x), from x= a to x= b. The word "perimeter" is reserved for closed geometric figures.
 

1. How do you calculate the perimeter of a region using integrals?

To calculate the perimeter of a region using integrals, you first need to find the equation of the curve that represents the boundary of the region. Then, you can use the integral of the function to find the length of the curve. Finally, you can add up all the lengths of the curves to find the total perimeter of the region.

2. What is the difference between using integrals and traditional methods to calculate perimeter?

The traditional method of calculating perimeter involves breaking the region into smaller shapes, such as rectangles or triangles, and adding up their lengths to find the total perimeter. Using integrals, on the other hand, allows for a more precise calculation by taking into account the curves and irregular shapes of the region.

3. Can integrals be used to calculate the perimeter of any region?

Yes, integrals can be used to calculate the perimeter of any region, as long as the boundary of the region can be represented by an equation or a set of equations. However, for more complex regions, the calculation may be more difficult and require advanced techniques.

4. Are there any practical applications of calculating perimeter using integrals?

Yes, calculating the perimeter of a region using integrals has several practical applications in fields such as engineering, physics, and geometry. For example, it can be used to determine the amount of fencing needed for a particular area or to calculate the surface area of complex objects.

5. Can integrals also be used to calculate the perimeter of three-dimensional objects?

Yes, integrals can be used to calculate the perimeter of three-dimensional objects by finding the surface area of the object. This can be done by breaking the three-dimensional object into smaller surfaces and using integrals to find the area of each surface, then adding them up to find the total surface area.

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