Simpson's RuleDate: 04/12/97 at 16:11:25 From: Adam Subject: Using Simpson's Rule to find volume Hi, Recently at school we came across a question which asks you to find the volume when a curve is rotated about the x-axis, using Simpson's rule. Could you please solve it and include a worked solution and theory behind your answer? Q. Find the volume of the solid formed if the curve y = cos -1(x) is rotated about the x-axis between x=0 and x=1 (use Simpson's Rule with 3 function values). Note: y = cos -1(x) means y = cos inverse of x. (cos to the power of -1) Thanks in advance. Date: 04/13/97 at 11:51:05 From: Doctor Anthony Subject: Re: Using Simpson's Rule to find volume The volume of revolution is given by INT[pi.y^2.dx]. So to use Simpson's Rule, we use the usual formula, but with ordinates y^2 instead of y, and also multiply by pi. The curve is the arccos curve - also denoted by cos^(-1)(x). With only three function values we have x=0, x=1/2 and x=1 The table of values is as shown below: x y = cos^(-1)(x) y^2 --------------------------------- 0 pi/2 2.4674 = y0 0.5 1.047 1.0966 = y1 1 0 0 = y2 Simpson's Rule then gives: (using h = 1/2 and the factor pi in front). V = pi/6[y0 + 4y1 + y2] = pi/6[2.4674 + 4 x 1.0966 + 0] = pi/6[6.8538] = 3.5886. -Doctor Anthony, The Math Forum Check out our web site! http://mathforum.org/dr.math/ |
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