EGR 599 Advanced Engineering Math II _____________________ LAST NAME, FIRST Problem set #8 1. Minimize the integral I= /2 0 dy 2 2 y 2 y dx with y(0) = 0, y(/2) = 0 dx Evaluate I. 2. Determine the path that minimizes the time that a particle will take to reach some lower point P(x = 4) along a flexible, frictionless wire under the force of gravity if it is released from rest at point a (x = 1, y = 0) as shown in Figure 1. 1 4 a x ds P y Figure 1 Find the path that minimizes the time from a to P. 3. Find the minimum area of a soap film supported by two rings of radius a arranged parallel to each other with their centers a distance 2b apart on an axis normal to the rings as shown. Calculate the numerical value with a = 1 and b = 0.4. y ds a 2b y x 4. (McQuarrie P. 20.2-4) Derive and solve Lagrange’s equation of motion for a mass falling vertically under the influence of gravity. At t = 0, x = x0, v = v0. 5. Of all parabolas which pass through the point (0, 0) and (1, 1), determine that one which, when rotated about the x axis, generates a solid of revolution with least possible volume between x = 0 and x = 1. [Notice that the equation may be taken in the form y = x + cx(1 x), where c is to be determined.] Ldx r dx 6. Use the formula = and the trial function (x) = c1x(1 x) + c2x2(1 x) to estimate 2 the smallest eigenvalue in equation d 2u = u dx 2 with u(0) = u(1) = 0. Hint: Use the definition of a beta function B(n, m) = 1 x 0 n 1 (1 x) m1 dx = p ( x ) y ' ( x ) q( x ) y r( x) y ( x)dx b 7. Use the formula = ( n ) ( m ) ( n m ) 2 a b 2 ( x ) dx and the trial function (x) = c1x(1 x) + 2 a c2 x2(1 x) to estimate the smallest eigenvalue in equation d 2u = u dx 2 with u(0) = u(1) = 0.