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GRE Math Subject Test Sample Question #1 [closed]

asked 2010-10-17 09:33:15 -0500

ccanonc gravatar image

updated 2015-01-14 06:58:03 -0500

FrédéricC gravatar image

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:

alt text

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:

alt text

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Closed for the following reason the question is answered, right answer was accepted by ccanonc
close date 2010-11-15 14:41:51

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Should we use the plot option `aspect_ratio=1` here to make the semicircle appear more circular?

Mitesh Patel gravatar imageMitesh Patel ( 2010-10-17 11:55:20 -0500 )edit

Good point Mitesh. Is there a reason it's not the default?

ccanonc gravatar imageccanonc ( 2010-10-17 11:59:05 -0500 )edit

I think this is to avoid squashed plots, e.g., `parametric_plot((100*cos(x), sin(x)), (x, 0, pi), aspect_ratio=1)`.

Mitesh Patel gravatar imageMitesh Patel ( 2010-10-17 12:19:44 -0500 )edit

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 ;-)

ccanonc gravatar imageccanonc ( 2010-10-17 12:25:21 -0500 )edit

I think it would be a nice feature if all plot docstrings sucked in a note about useful common kwargs like aspect_ratio.

ccanonc gravatar imageccanonc ( 2010-10-17 12:27:27 -0500 )edit

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answered 2010-10-17 10:35:33 -0500

Mike Hansen gravatar image

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
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Comments

Nice formatting!

ccanonc gravatar imageccanonc ( 2010-10-17 10:44:45 -0500 )edit

The MathJax "bar of inclusion" on the sqrt doesn't have a terminating down stroke before the "dt". ;p Outrageous!

ccanonc gravatar imageccanonc ( 2010-10-17 21:20:14 -0500 )edit

Does LaTex do that downstroke? Mine doesn't usually.

kcrisman gravatar imagekcrisman ( 2010-10-18 04:37:56 -0500 )edit

I was just joking. Mike's post is nearly flawless as usual.

ccanonc gravatar imageccanonc ( 2010-10-18 05:12:12 -0500 )edit

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Asked: 2010-10-17 09:33:15 -0500

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Last updated: Oct 17 '10