Proving a Statement about a Regular Curve: R of Area k

In summary, the conversation discusses the use of Green's theorem to prove that if C is a simple regular curve enclosing a region R of area k, then the integral of [a1x + a2y + a3, b1x + b2y + b3] over C is equal to (b1 - a2)k. The conversation also includes a link for the topic being discussed.
  • #1
Fernando Revilla
Gold Member
MHB
631
0
I quote a question from Yahoo! Answers

If C is a simple regular curve that encloses a region R of area k. Prove that if ai, bi are constants
∫(c) [a1x + a2y + a3, b1x + b2y + b3]dα = (b1 - a2)k. thanks

I have given a link to the topic there so the OP can see my response.
 
Mathematics news on Phys.org
  • #2
Using the Green's theorem: $$\int_C(a_1x+a_2y+a_3,b_1x+b_2y+b_3)d\alpha=\iint_R\left(\frac{\partial Q}{\partial x}-\frac{\partial P}{\partial y}\right)dxdy\\=\iint_R(b_1-a_2)dxdy=(b_1-a_2)\iint_Rdxdy=(b_1-a_2)k$$
 

FAQ: Proving a Statement about a Regular Curve: R of Area k

What is a regular curve?

A regular curve is a continuous curve with no sharp turns or discontinuities. It is also known as a smooth curve.

How do you prove a statement about a regular curve?

To prove a statement about a regular curve, you need to use mathematical techniques and principles such as calculus, geometry, and algebra. You may also need to use specific theorems and formulas related to regular curves.

What does "R" represent in the statement "R of Area k"?

In this context, "R" represents the regular curve in question, while "Area k" represents the area enclosed by the regular curve.

Can any statement be proven about a regular curve?

Not necessarily. The statement must be mathematically sound and logically valid in order to be proven. Additionally, it must be possible to apply the appropriate mathematical techniques to the regular curve in question in order to prove the statement.

Are there any real-life applications for proving statements about regular curves?

Yes, there are many real-life applications for proving statements about regular curves. For example, these principles are used in physics, engineering, and architecture to analyze and design structures such as bridges, roller coasters, and buildings. They are also used in computer graphics and animation to create smooth and realistic curves.

Similar threads

MHB Area Enclosed by One Petal of a Rose Curve: r = 8sin7θ
Replies
1
Views
9K
MHB Area bounded by 3 curves: Help with Problem Solving
Replies
1
Views
6K
MHB Proving $\displaystyle\int_C \vec{F}\cdot d\vec{r} = 0$ with Closed Curve
Replies
6
Views
2K
MHB Joko's question at Yahoo Answers (r(t) perpendicular to r'(t))
Replies
1
Views
2K
Prove that every right triangular region is measurable and its area is 1/2bh
Replies
5
Views
4K
Every open set contains both open and closed cell.
Replies
3
Views
4K
Schools How well does this course prepare me for college math?
Replies
10
Views
4K

Hot Threads

Recent Insights

Back
Top