13.7 problem 8

Hi Dr. Taylor,
I was wondering if you might help me get started on this
problem. I'm having some difficulty figuring out the parameterization as
well as the bounds.
Thank you,
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First of all the cylinder x^2+y^2=16 has radius 4 and is parallel to the z axis.  Then the planes x+y+z=3 and x+y+z=7  are parallel to each other, and you can solve these for z  as z=3-x-y and
z=7-x-y.  In particular you can get from first plane to the other by increasing z by 4.  Then points on the cylinder can be specified by writing x=4 cos(u), y=4 sin(u), while the intersection of the cylinder with the first plane also specifies z=3-4 cos(u)-4 sin(u) while the intersection with second plane specifies instead that z=7-4 cos(u)-4 sin(u).  You can put this all together to get a parameterization of the space on the cylinder between the two planes by saying
x=4 cos(u), y=4 sin(u), z=3+v-4 cos(u)-4 sin(u)
where 0≤u≤2π and 0≤v≤4.