Wednesday

August 20, 2014

August 20, 2014

Posted by **mark** on Monday, September 22, 2008 at 7:26pm.

Starting from the known expression for the length of a plane curve in Cartesian coordinates and the equation x = rcos@, y = rsin@, obtain an integral expression in terms of the function g and the numbers a and b for the length of the given curve.

How would I do this question. Thanks.

- calculus -
**Count Iblis**, Monday, September 22, 2008 at 9:14pmIf ds is a length element, then:

ds^2 = dx^2 + dy^2

In polar coordinates:

x = r cos(theta)

y = r sin(theta)

Using Leibnitz rule:

dx = cos(theta)dr -sin(theta) r dtheta

dy = sin(theta)dr +cos(theta) r dtheta

ds^2 = dx^2 + dy^2 =

[cos^2(theta) + sin^2(theta)] dr^2 +

[cos^2(theta) + sin^2(theta)] r^2 dtheta^2 +

2 [sin(theta)cos(theta) - cos(theta)sin(theta)] r dr dtheta.

The contents of first two square brackets are equal to one, the last is zero. So, we have:

ds^2 = dr^2 + r^2 dtheta^2

You could have imediately written down this equation. At any point along the curve, dr is a length element in the radial direction and r dtheta is the length element in the tangential direction, which is orthogonal to the radial direction. So, by Pythagoras' theorem, ds^2 is the sum of the suares of the two length elements.

It then follows that:

ds = sqrt[dr^2 + r^2 dtheta^2] =

sqrt[(dr/dtheta)^2 + r^2] dtheta

Integrating both sides over theta from theta = a to theta = b then gives the curve length.

**Related Questions**

Calculus - Consider a plane curve which is described in polar coordinates (r...

calculus - Consider a plane curve which is described in polar coordinates (r...

Pre-calculus; urgent ! Please help ! - 1.Graph the polar equation r=3-2sin(theta...

Calculus please help ! - 1.Graph the polar equation r=3-2sin(theta) 2. Find the ...

Math ! Please Help ! - 1.Graph the polar equation r=3-2sin(theta) 2. Find the ...

Math (Calculus) - Given any Cartesian coordinates, (x,y), there are polar ...

calculus - Find the area of the region lying inside the polar curve r = 2(1 &#...

calculus - Find the area of the region lying inside the polar curve r = 2(1 &#...

MATH - (a) Transform the expression (x − a)^2 + y^2 = a^2 into polar ...

calculus - find the area of the region bounded by the polar curve r=sqrt(6ln(...