2. 136.170: Test #2
Solutions
Name:_____________________ Student Number: _______
[6] 1.Differentiate:
.
By the Fundamental Theorem of Calculus and by the chain rule:
2. Compute the following integrals:
[6] (a)
[6] (b)
(a) Use the substitution
.Tha gives us 
so that 
. Notice also that as t changes from 0 to 2, 
changes from 0 to 4. We get 
.
(b)
and use the subsitution 
. Then 
and we have 
.
Alternative: start wit the substitution
.
[7] 3. Sketch (roughly) the region bounded by the curves
and 
and then find its area.
The first coordinates of the two intersection points are found by solving
and 
. We find that x=0 and x=1 for the two solutions. So, the area is 
.