Line Integral on R2 Curve in Polar Coordinates

In summary, the conversation discusses a line integral in polar coordinates, specifically the integral from θ1 to θ2 of f(r*cosθ, r*sinθ) multiplied by the square root of r^2 + (dr/dθ)^2, with x=cosθ and y=sinθ as the parametrization for the curve. The conversation also explores the meaning of dr/dθ and its graphical representation in relation to the line integral.
  • #1
Mr.MaestrO
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Homework Statement


Consider a curve in R2 given in polar coordinates r=r(θ) for θ1<= θ <= θ2. Show that the line integral is equal to the integral from θ1 to θ2 of f(r*cosθ, r*sinθ) sqrt (r^2 + (dr/dθ)^2) dθ


Homework Equations


x= cos θ, y= sin θ


The Attempt at a Solution


I understand that the curve does not necessarily have to be linear. So in Polar Coordinates we can let x=r*cosθ and y=r*sinθ as our parametrization for the curve. Line integral tells us that it is equal to the integral of the region (θ1 to θ2) of f(g(θ)) multiply by the magnitude of g(θ). g(θ) = (r*cos θ, r*sin θ), and the square root term is the magnitude of g(θ). My question is, where did the dr/dθ come from? What does it mean graphically?

Thanks!
 
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  • #2
Welcome to PF!

Hi Mr.MaestrO! Welcome to PF! :smile:

(have a square-root: √ and try using the X2 icon just above the Reply box :wink:)

Hint: arc-length (ds)2 = (dx)2 + (dy)2 = … ? :smile:
 

1. What is a line integral on R2 curve in polar coordinates?

A line integral on R2 curve in polar coordinates is a mathematical concept used in multivariable calculus to calculate the total change of a function along a given curve in the polar coordinate system. It takes into account both the length of the curve and the values of the function along the curve.

2. How is a line integral on R2 curve in polar coordinates different from a regular line integral?

A line integral on R2 curve in polar coordinates differs from a regular line integral in that it takes into account the curvature of the curve in the polar coordinate system. This allows for a more accurate calculation of the total change of the function along the curve.

3. What is the formula for calculating a line integral on R2 curve in polar coordinates?

The formula for calculating a line integral on R2 curve in polar coordinates is given by ∫f(r(t))|r'(t)|dt, where f is the function being integrated, r(t) represents the curve in polar coordinates, and |r'(t)| is the length of the curve.

4. What is the significance of using polar coordinates in line integrals?

Using polar coordinates in line integrals allows for a more efficient and accurate calculation of the total change of a function along a given curve. It also simplifies the calculation process by taking into account the curvature of the curve.

5. Can a line integral on R2 curve in polar coordinates be applied to real-world problems?

Yes, line integrals on R2 curve in polar coordinates have many real-world applications, such as calculating work done by a force along a curved path or finding the flux of a vector field through a curved surface. It is a useful tool in physics, engineering, and other fields that involve the analysis of curved paths and surfaces.

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