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Expert: - 3/16/2010


Question
Scott,
How do I calculate the area of a ring with a triangular cross section (base .5", Height 1") The triangle sweeps out a ring with a diameter of 3"

Answer
It sounds like you're talking about swinging a triangle around in a ring.
The outside of the ring has a diameter of 3", so the radius is 1.5".
That means the inside of the triangle is 1.5 - 0.5 = 1.

This will generate a integral in polar coordinates.
Here is an article on polar coordinates (cut and paste in your search box):
http://mathworld.wolfram.com/PolarCoordinates.html

Remember that integrals in polar coordinates are ∫∫ f(r,Θ)r dr dΘ.
In this case, f(r,Θ) = 2r and the limits of integration are for r going from 1 to 1.5.
The values of Θ go from 0 to 2π.

To do ∫∫ 2r² dr dΘ, you get ∫ 2r³/3 dΘ with r going from 1.5 down to 1.
This means it is ∫(1.5³-1³)/2 dΘ = ∫(3.375 - 1)/2 dΘ = ∫2.375 dΘ.
Now that integrates to 2.375Θ for Θ going from 2π down to 0.
This evaluates to 2.375π.

This can be done in rectangular coordinates, but it is a lot harder.
I believe it leads to a substitution that takes it to polar coordinates,
but it is seen as a trig substitution.  Doing polar at first avoids that step.

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Scott A Wilson

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I can answer any question in general math, arithetic, discret math, algebra, box problems, geometry, filling a tank with water, trigonometry, pre-calculus, linear algebra, complex mathematics, probability, statistics, and most of anything else that relates to math. I can even tell you it takes me over 2,000 steps to go a mile, but is that relevant?

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Experience in the area; I have tutored people in the above areas of mathematics for almost two years in AllExperts.com. I have tutored people here and there in mathematics since before I received a BS degree almost 25 years ago. In just two more years, I received an MS degree as well, but more on that later. I tutored at OSU in the math center for all six years I was there. Most students offering assistance were juniors, seniors, or graduate students. I was allowed to tutor as a freshman. I tutored at Mathnasium for well over a year. I worked at The Boeing Company for over 5 years. I received an MS degreee in Mathematics from Oregon State Univeristy. The classes I took were over 100 hours of upper division credits in mathematical courses such as calculus, statistics, probabilty, linear algrebra, powers, linear regression, matrices, and more. I graduated with honors in both my BS and MS degrees. Past/Present Clients: College Students at Oregon State University, various math people since college, over 7,500 people on the PC from the US and rest the world.

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My master's paper was published in the OSU journal. The subject of it was Numerical Analysis used in shock waves and rarefaction fans. It dealt with discontinuities that arose over time. They were solved using the Leap Frog method. That method was used and improvements of it were shown. The improvements were by Enquist-Osher, Godunov, and Lax-Wendroff.

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Master of Science at OSU with high honors in mathematics. Bachelor of Science at OSU with high honors in mathematical sciences. This degree involved mathematics, statistics, and computer science. I also took sophmore level physics and chemistry while I was attending college. On the side I took raquetball, but that's still not relevant.

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I earned high honors in both my BS degree and MS degree from Oregon State. I was in near the top in most of my classes. In several classes in mathematics, I was first. In a class of over 100 students, I was always one of the first ones to complete the test. I graduated with well over 50 credits in upper division mathematics.

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