# here ```EGR 599
_____________________
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.]
 Ldx
 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) m1 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.
```