4. 136.170: Test #2
Solutions
[6] 1.Differentiate:
. (Hint: start with the Product Rule of differentiation!)
2. Compute the following integrals:
[6] (a)
[6] (b)
(a) For
we use the substitution 
. We see that 
, that 
and that as x changes from 0 to 1, 
changes from 1 to 2. So, 
.
(b) Use
. So 
and 
.
[7] 3. Sketch (roughly) the region bounded by the curves
and 
and then find its area.
The second coordinates of the two intersection points are found by solving
and 
. We find that y=0 and y=1 for the two solutions. So, the area is 
.