Calculus II 
Test #2
March 20, 2002
Name____________________    
R.  Hammack 
Score ______

(1) Find the area of the region bounded by the curves y = tan(x),   y = 1,  and  x = 0.

This problem is going to involve finding an antiderivative of tan(x), so let's get that out of the way first.
[Graphics:Images/temp_gr_1.gif][Graphics:Images/temp_gr_2.gif][Graphics:Images/temp_gr_3.gif]
Making the substitution u = cos(x), we get du = -sin(x)dx.
The above integral becomes [Graphics:Images/temp_gr_4.gif].

 

Now we can get down to business. Notice that the curves y = tan(x) and y = 1 intersect where x = π/4,
and y = 1 is the top function and y = tan(x) is the bottom function. The area we seek is thus


[Graphics:Images/1S02Dsol_gr_44.gif] [ x +ln |cos x| [Graphics:Images/1S02Dsol_gr_45.gif]= π/4+ln cos π/4 = π/4 + ln (1/[Graphics:Images/1S02Dsol_gr_46.gif]) square units


(2) Consider the region contained under the graph of [Graphics:Images/1S02Dsol_gr_56.gif] between x = 0 and x = 4.
The region is revolved around the x-axis. Find the volume of the resulting solid.
Finding volume by cross-sectional area,
[Graphics:Images/1S02Dsol_gr_57.gif][Graphics:Images/1S02Dsol_gr_58.gif][Graphics:Images/1S02Dsol_gr_59.gif]= 8π cubic units


(3) Consider the region contained under the graph of [Graphics:Images/1S02Dsol_gr_60.gif]between x = 0 and x = [Graphics:Images/1S02Dsol_gr_61.gif].
The region is revolved around the y-axis. Find the volume of the resulting solid.
Finding volume by shells,
[Graphics:Images/1S02Dsol_gr_62.gif][Graphics:Images/1S02Dsol_gr_63.gif][Graphics:Images/1S02Dsol_gr_64.gif]π[Graphics:Images/1S02Dsol_gr_65.gif]=π(-cos π + cos 0) = 2π cubic units
u = [Graphics:Images/1S02Dsol_gr_66.gif]
du = 2x dx






(4)
Consider the curve [Graphics:Images/temp_gr_9.gif] for [Graphics:Images/temp_gr_10.gif]. Find the area of the surface that results when this curve is revolved around the x-axis.

[Graphics:Images/temp_gr_11.gif][Graphics:Images/temp_gr_12.gif]  
[Graphics:Images/temp_gr_13.gif] [Graphics:Images/temp_gr_14.gif] [Graphics:Images/temp_gr_15.gif]  

[Graphics:Images/temp_gr_16.gif] [Graphics:Images/temp_gr_17.gif]  [Graphics:Images/temp_gr_18.gif]  

[Graphics:Images/temp_gr_19.gif]square units



(5)
A variable force pushes an object  3 feet in a straight line. When the object is x feet from its starting point, the force on the object is  [Graphics:Images/temp_gr_20.gif] pounds. How much work is done in moving the object 3 feet?

[Graphics:Images/temp_gr_21.gif][Graphics:Images/temp_gr_22.gif]foot pounds