Science:Math Exam Resources/Courses/MATH101/April 2005/Question 06 (a)
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Question 06 (a) 

A tree trunk is 60 feet long and has circular cross sections. The diameters in feet measured at 10foot intervals starting at the bottom of the tree are 10, 8, 7, 6, 5, 4, and 3. The volume of the tree, in cubic feet, is computed using the formula where is the crosssectional area of the trunk, in square feet, feet above the bottom of the tree. (a) Using a Simpson's Rule approximation for , give an estimate for the volume of wood in the tree trunk. (Do not simplify your answer.) 
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you? 
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint. 
Hint 1 

To get the volume of the tree you will need to write a Simpson's Rule sum involving the cross sectional area of the trunk. Look closely at what you are given when evaluating the sum. 
Hint 2 

The formula for the Simpson's rule is given by
where as always is an even integer. 
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. This is a straightforward application of Simpson's Rule to the integral Notice that
By Simpson's Rule, the integral can be approximated by the sum Remembering the given information about the diameter of the tree trunk at various heights, , we make the appropriate substitutions to determine the cross sectional area at each height: Plugging into our sum we get the approximate volume for the tree trunk. 