Ah that makes sense. Is there a reason how you knew that the limits in the question would be associated with the upper half of the curve and not the bottom?
Ah that makes sense. Is there a reason how you knew that the limits in the question would be associated with the upper half of the curve and not the bottom?
absolutely the values of theta which produce the top half lie between -pi/3 and 2pi/3 They are even marked in the picture
absolutely the values of theta which produce the top half lie between -pi/3 and 2pi/3 They are even marked in the picture
I can see that they are marked in the picture, but what I'm asking is how you knew that, when you were working the limits out, they would be associated with the top half?
I can see that they are marked in the picture, but what I'm asking is how you knew that, when you were working the limits out, they would be associated with the top half?
because pi/2 which also is marked in the picture lies on the top half and happens to lie between -pi/3 and 2pi/3
What I'm saying is why isn't the area calculated this when you integrate between A and E. If you integrated this using Cartesian coordinates (ignoring the bottom half of the ellipse) between A and E you would calculate the area shown above wouldn't you. But when switching to parametric integration it doesn't seem to be the same. I thought that the only difference between parametric and normal Cartesian integration was that you use the t value for the corresponding c values as the limits. And you integrate with respect to t (and of course multiplying brought by dx/dt). It seems here that there is something fundamentally flawed about how I'm thinking of it.
What I'm saying is why isn't the area calculated this when you integrate between A and E. If you integrated this using Cartesian coordinates (ignoring the bottom half of the ellipse) between A and E you would calculate the area shown above wouldn't you. But when switching to parametric integration it doesn't seem to be the same. I thought that the only difference between parametric and normal Cartesian integration was that you use the t value for the corresponding c values as the limits. And you integrate with respect to t (and of course multiplying brought by dx/dt). It seems here that there is something fundamentally flawed about how I'm thinking of it.