The trough has the cross section of a parabola and a tube
running the length of the parabola's focus heats up when
exposed to sunlight.  Here is the design and implementation of
such a solar collector.
See Nested Parabolas on another page on this site.

Use Nested Parabolas to find the curve that accounts for 1/4 inch
plywood backing as detailed in
another page in this site.  This is the
cut in the frame that results in the precise parabola surface facing the

Note that in calculating the arclength, not all math programs are the
same.  There was a 1/2" error using Graph, so I resorted to using a
Spreadsheet.  I used trial and error to find what focus would accept a
48 inch wide sheet of 1/4 inch plywood curved into a shallow
parabola, on a 48 inch wide 1/2 inch thick plywood frame with 2
inches to spare on each side.  That is, the 4' wide 1/4" plywood must
be bent minimally into a parabola that is 44" wide at its top, so that
the frame can be cut out of 4' wide 1/2" plywood with structural
At x=+/-21.929", y=12.266" defines the
ledge against which the 4' width of 1/4"
plywood is hooked into place.  The rest
of the x values are given in 2" intervals.  
The y values for each interval were
calculated using experimentation on a
spreadsheet.  The values would be
good candidates for Newton's Method
for more precise results.  Note that 4" is
added to each y value to accommodate
for the frame.  These points are plotted
on large graph paper and the points
connected with a flexible ruler to
describe the curve.  These are the
points that describe the lower curve
that is cut into the frame so that after
the 1/4" plywood is applied it results in
the exact depiction of the parabola.  A
thick layer of reflective Mylar is then
glued to the surface of the plywood for
the economical result.