Hi Göksu,
I understand that you would like to differentiate the equations of a line segment, a circle, and an ellipse from each other so that you could parameterize each of them with their respective formula. You can do so by analysing the equation of the curve provided by the user.
1. Line Segment:
A line segment can be represented by the equation of a straight line in the form 
, where m is the slope and c is the y-intercept. To identify a line segment, check if the equation follows this linear form.
, where m is the slope and c is the y-intercept. To identify a line segment, check if the equation follows this linear form.If it does, you can parametrize it using the parameter t,
Here, (x1, y1) and (x2, y2) are the coordinates of the starting and ending points of the line segment, respectively. The parameter t varies between 0 and 1, representing the proportion of the line segment's length.
2. Circle:
A circle can be represented by the equation 
, where (h, k) represents the centre coordinates and r is the radius. To identify a circle, check if the equation follows this form.
, where (h, k) represents the centre coordinates and r is the radius. To identify a circle, check if the equation follows this form.If it does, you can parametrize it using the parameter θ,
Here, θ represents the angle around the circle (0 ≤ θ ≤ 2π)
3. Ellipse:
An ellipse can be represented by the equation 
, where (h, k) represents the centre coordinates, and a and b are the semi-major and semi-minor axes, respectively. To identify an ellipse, check if the equation follows this form.
, where (h, k) represents the centre coordinates, and a and b are the semi-major and semi-minor axes, respectively. To identify an ellipse, check if the equation follows this form.If it does, you can parametrize it using the parameter θ,
Here, θ represents the angle around the ellipse (0 ≤ θ ≤ 2π)
Note that these parametrizations assume a specific orientation and starting point for each curve. Adjustments may be required depending on the desired orientation and starting point.
Once you have determined the type of curve based on its equation form, you can use the appropriate parametrization to calculate the line integral.
