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Volumes by Integration

  • Thread starter planauts
  • Start date
  • #1
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Hi, I am having a lot of trouble with this problem.

Homework Statement



A lumberjack is preparing to cut down a large maple tree. With a chain saw, he makes a horizontal cut exactly halfway through the trunk, and then makes a second cut at 45 degrees, meeting the first cut along the diameter. Determine the volume of the wedge of wood cut out if the diameter is 0.9m.

Homework Equations


Hint: Take the element of integration to be a vertical slice parallel to the thin edge of the wedge.
ANSWER: 0.06075 m^3

The Attempt at a Solution


[PLAIN]http://img232.imageshack.us/img232/8066/80795462.gif [Broken]

dV = (area)*(dy)
I think the area would be: r(theta) = r(pi/4) = (0.9)(pi/4)
But I am stuck after that...I don't even think I am on the right track....

Thanks for your help!
 
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Answers and Replies

  • #2
118
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Well, the idea is that to find the volume of the wedge, you should slice it up, express the area of each wedge in terms of variables, and then integrate that expression to give the volume of the whole wedge.
So, how are you going to slice up the wedge?
What you have written down so far doesn't make much sense. It seems as though you're trying to slice the wedge so that you get triangles. That makes sense. But not all the triangles you get are going to have the same area. How do you express the area of each slice in terms of what variables you're given?
It looks like you tried to find the area of one triangle by using r*theta, but that's not even the formula for the area of a triangle, which is 1/2(base)(ht).
 
  • #3
A better way to picture this is to first imagine half of a circle on the xy plane. Then place triangles on the circle perpendicular to the plane with the right angle of the triangle sitting on the curve of the circle. Now the length is bounded by this semicircle so the length changes from 0->0.45->0 as you move the triangle from the edge of the circle to the middle to the other edge. Hope this helps.
 

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