What is the length of the parametric curve: f(t) = (x,y) = (cos(t),sin(t)), for t in [0,Pi] ?
Update: Mike got the answer lightning-/mike-fast. Here's the output of parametric_plot() below:
Update 2: Per Mitesh's suggestion, I called parametric_plot() with keyword argument "aspect_ratio=1"; the image below looks more round with a 1:1 aspect ratio:
The curve is a semicircle with radius 1 -- it's length is $\pi$. You can use the arc-length formula to get this
$$ \int_0^\pi \sqrt{cos(t)^2+sin(t)^2} dt = \int_0^\pi 1 dt = \pi $$.
Finally, in Sage:
sage: parametric_plot((cos(x), sin(x)), (x, 0, pi))
sage: integrate(sqrt(cos(x)^2+sin(x)^2),x, 0, pi)
pi
Asked: 2010-10-17 16:33:15 +0100
Seen: 1,159 times
Last updated: Oct 17 '10
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Should we use the plot option `aspect_ratio=1` here to make the semicircle appear more circular?
Good point Mitesh. Is there a reason it's not the default?
I think this is to avoid squashed plots, e.g., `parametric_plot((100*cos(x), sin(x)), (x, 0, pi), aspect_ratio=1)`.
Maybe some library, say GraphicsMagick (a subset of ImageMagick) could intelligently use a heuristic to pick a better default aspect ratio? .... My 5.0 wishlist ;-)
I think it would be a nice feature if all plot docstrings sucked in a note about useful common kwargs like aspect_ratio.