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January 12th, 2012, 03:26 PM  #1 
Member Joined: Aug 2011 From: Alabama Posts: 93 Thanks: 0  Estimated Volume of a Solid Hemisphere
I wasn't in class today when we were talking about spherical approximation. The question says: To estimate the volume of a solid hemisphere of radius 4, imagine its axis of symmetry to be the interval [0,4] on the axis. Partition [0,4] into 8 subintervals of equal length and approximate the solid with cylinders based on the circular cross sections of the hemisphere perpendicular to the xaxis at the subintervals' left endpoints. [attachment=0:37qlb9pm]scan0034.jpg[/attachment:37qlb9pm] (a)Find the sum of of the volumes of the cylinders. Do you expect to overestimate ? Give reasons for your answer. (b) Express as a percentage of to the nearest percent. 
January 12th, 2012, 04:24 PM  #2 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,206 Thanks: 513 Math Focus: Calculus/ODEs  Re: Estimated Volume of a Solid Hemisphere
What they want you to do is revolve each of the rectangles about the xaxis 180°, forming a stack of cylinders, which will approximate the volume of a hemisphere. The radius of each cylinder will be equal to y(x) for x = 0.5k, where k is in {0,1,3,...,7}. Thus: We should expect this to be an overestimate based on the fact that all of the rectangles extend beyond the circle. 
January 12th, 2012, 07:11 PM  #3  
Member Joined: Aug 2011 From: Alabama Posts: 93 Thanks: 0  Re: Estimated Volume of a Solid Hemisphere
Thanks for the reply, but I'm totally in the dark. Do you know of a website or something that could give me a rundown on how this works. I can't seem to find one. Quote:
 
January 12th, 2012, 07:49 PM  #4 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,206 Thanks: 513 Math Focus: Calculus/ODEs  Re: Estimated Volume of a Solid Hemisphere
Let me try to explain it better... Imagine each of the rectangles is the crosssection of a cylinder whose height is 1/2 unit. Begin with the leftmost rectangle. It is 8 units in height (2y(0)), so the radius of that cylinder would be 4 units. The volume of a cylinder is With h = 1/2 and r = 4, its volume would be Continuing, we see that for each cylinder, we have and for the nth cylinder, x = (n  1)(1/2). Thus, the volume of the nth cylinder is: Let n be the natural numbers 18, then add the volumes all together (you may choose to do it differently, this is how I would do it rather than manually computing 8 separate volumes to save some time): So now we find: 
January 12th, 2012, 08:12 PM  #5 
Member Joined: Aug 2011 From: Alabama Posts: 93 Thanks: 0  Re: Estimated Volume of a Solid Hemisphere Thank you! Unfortunately, I will have to do all 8 manually because the teacher is strict about things like that, but I appreciate it! To find x, you were solving for the midpoint, correct?

January 12th, 2012, 08:15 PM  #6 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,206 Thanks: 513 Math Focus: Calculus/ODEs  Re: Estimated Volume of a Solid Hemisphere
No, I was using the left endpoints of each subinterval. The first subinterval is from 0 to 1/2, so I used 0, the second subinterval is 1/2 to 1, so I used 1/2 and so on. The last subinterval is from 7/2 to 4 so I used 7/2. Using the midpoint would be more accurate, but trust me, you will be introduced to that method as well as several others. Approximating definite integrals is a fascinating and rich subject. 
January 12th, 2012, 08:42 PM  #7 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,206 Thanks: 513 Math Focus: Calculus/ODEs  Re: Estimated Volume of a Solid Hemisphere
Your teacher might just be impressed with you being able to sum a sequence of squares: (I can show you how to derive this, or prove it using induction, or both, if you're interested) In my first post above, I wrote: We may rewrite this as: Using the above formula, this becomes: (15)}{6}\)" /> 
January 12th, 2012, 08:48 PM  #8 
Member Joined: Aug 2011 From: Alabama Posts: 93 Thanks: 0  Re: Estimated Volume of a Solid Hemisphere
I kind of already did all 8, but for the sake of knowledge, I wouldn't mind you explaining how it works.

January 12th, 2012, 08:52 PM  #9 
Member Joined: Aug 2011 From: Alabama Posts: 93 Thanks: 0  Re: Estimated Volume of a Solid Hemisphere
In fact, if I could show my teacher a proof, he would definitely let me use it, so I's love to understand this 
January 12th, 2012, 09:49 PM  #10 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,206 Thanks: 513 Math Focus: Calculus/ODEs  Re: Estimated Volume of a Solid Hemisphere
Great! First, I'll prove it using induction, which requires having the formula: First, we check it for the base case of n = 0: this is true, so we assume the following is true: Now, we add to both sides: We have derived from thereby completing the proof by induction. Now, to derive the formula, one may begin with the nonhomogeneous recurrences and complete the process of symbolic differencing until a homogeneous relation is obtained: (1) (2) Subtracting (1) from (2) gives: (3) (4) Subtracting (3) from (4) gives: (5) (6) Subtracting (5) from (6) gives: Now we have a homogeneous relation whose associated auxiliary equation is: With the repeated root of r = 1 of multiplcity 4, we know then that the closed form will take the form: We could also have gotten to this point by observing a constant difference in the 3rd degree: n.....S_n.....?S.....?(?S).....?(?(?S)) 0......0.................................... 1......1........1........................... 2......5........4........3.................. 3......14.......9........5...........2.... 4......30.......16.......7...........2.... This tells us will be a cubic polynomial as well. Next, we use initial conditions to determine the coefficients : thus we know giving us the following 3X3 system: Solving the system, we find: and thus: 

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