14.6 Project Archimedes' Floating Paraboloid

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Of course, you will need to insert the z- and y-limits corresponding to your own values of a and b. We recommend that — as an experience worth the effort — you evaluate your three multiple integrals manually, taking the solution of Example 5 in the text as a model. That is, • First integrate (trivially) with respect to x. • Then integrate (easily) with respect to z, leaving you with an integral involving the 3 /2-power of a quadratic in y. 2 2 • Complete the square to express this quadratic in the form k − u where k = ( a+ b)/2 and u = y−( b−a)/2. • Make the trigonometric substitution u = ksin θ. • Finally, use (as necessary) the tabulated integral of cos n u inside the back cover of the textbook. In the technology-specific sections of this project we illustrate the automated computer algebra calculation of the volume and centroid of T. Archimedes' Answers As indicated in the 3-dimensional figure on the preceding page, the (top) base of our oblique segment is an ellipse with major axis P1P2, minor axis Q1Q2, and center M. The 2-dimensional figure at the top of the next page shows the intersection of the segment with the vertical yz-plane. The vertex V of the oblique segment is the point of the paraboloid at which its tangent plane is parallel to its base. Archimedes showed that • The volume of the segment is 3/2 that of the inscribed ellipsoidal cone with the same vertex and base. • The centroid of the segment lies on its axis VM and is 2/3 of the way from V to M. For a worthy challenge, verify that your own volume and centroid results agree with these results of Archimedes. A worthy challenge for both you and your computer algebra system is to show that the region bounded below by the paraboloid 2 2 z = k( x + y ) and above by the plane z = my+ n has volume π ( m + 4 kn) V = 3 32k and centroid ( x, y, z ) with coordinates 2 2 2 2 2 π mm ( + 4 kn) 5m+ 8kn x = 0, y = , z = . 4 64k 12k 14.6 Project 4
Using Maple P1 y M 14.6 Project 5 C V P2 a b x Taking a = 1 and b = 2 for an example, we want to investigate the segment of a paraboloid that is bounded below by the paraboloid z = x 2 + y 2 and above by the plane z = y + 2. If we view this oblique segment looking inward along the x-axis, we see that it is bounded front-and-back by the surfaces defined on the yz-region that is bounded above by the line z = y + 2 and below by the parabola z = y 2 . This region lies over the interval –1 < y < 2 on the y-axis. plot({y+2, y^2}, y=-1..2, color=blue, thickness=2);
Page 1 and 2: 14.6 Project Archimedes' Floating P
Page 3: z = x 2 + y 2 and below by the plan
Page 7 and 8: Thus our oblique segment of a parab
Page 9 and 10: centroid = (2/3)*M + (1/3)*V; Using
Page 11 and 12: Archimedes' Way First we examine he
Page 13 and 14: Using MATLAB Taking a = 1 and b = 2
Page 15 and 16: Archimedes' Way First we examine he

14.6 Project Archimedes' Floating Paraboloid

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