Megan,

      Here is the 7th problem set in MATH 136:

    Individual Problem Set 7 -- From Stewart:

    * Section 5.10/14, 15, 22, 24, 29, 50, 62;
    * Section 6.1/6, 8, 15, 31;
    * Section 6.2/4, 5, 6, 16, 26, 32, 36.
    * Section 6.5/5, 8. 

    Of these, please grade the following problems:


     5.10/14 -- 1 point for correct set up with limit
                3 for value of integral from 1 to b (via u-substitution)
                1 for limit as b goes to infinity
                1 for "integral is convergent"

     5.10/29 -- 6 points as for 14 above. (Note: here the discontinuity
                at x = 1 is in the interior of the interval, so  
                only give full credit if they split the integration
                from 0 to b (b < 1) and a to 33 (a > 1), then 
                let b -> 1-  and a -> 1+


     5.10/62 -- 4 points:  2 for solving y = e^{-x^2} for x in terms
                of y.  2 for a reason why the integrals are equal.
                A picture is OK by itself if they show both regions
                of integration. A verbal explanation in terms of 
                inverse functions and the geometric operation of 
                reflecting the region for the x-integral across 
                the line y = x is also OK.  The solutions manual 
                doesn't say it that way, but that is really what 
                is going on.

     6.1/8  --  6 points:  2 for finding the intersection points,
                2 for setting up the integral correctly, 2 for
                evaluation of integral.  Only take off one point
                if they set it up OK but make an arithmetic mistake.
                Also, do not take off if they don't draw the picture
                with the "representative rectangle"

     6.2/4  --  6 points: Part credit: 1 for general set up with 
                disk cross sections by planes perpendicular to 
                y-axis, 1 for x = e^y, 2 for the correct integral, 
                2 for the value.  

     6.2/32 --  6 points:  I expect most students who do this one
                will develop a solution like the first one in the 
                manual, although the orientation of the axes may 
                be reversed (the frustum may be obtained by rotating 
                about the x-axis).  That is OK.   Don't take off 
                points if the volume is correct but unsimplified.  

     6.5/8  --  4 points for part a (1 for correct integral, 2 for 
                correct antiderivative x ln(x) - x -- either using parts
                or the table is OK), 1 for value

                2 points for part b

                2 points for picture in part c

     Note that this is a smaller number of problems than usual.  I 
     expect these will be hard to grade, and I would like you to 
     provide as much feedback as you reasonably can in cases where
     the students are seriously wrong or where they do not attempt
     a problem.  For the volume problems, for instance, in those
     cases, say what the cross-sections are (disks, washers, etc.)
     and give cross-section area and the integral used to compute 
     the volume.  Thanks!

                                             John Little