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.

Making the substitution u = cos(x), we get du = -sin(x)dx.
The above integral becomes .

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

[ x +ln |cos x| = π/4+ln cos π/4 = π/4 + ln (1/) square units

(2) Consider the region contained under the graph of 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,
= 8π cubic units

(3) Consider the region contained under the graph of between x = 0 and x = .
The region is revolved around the y-axis. Find the volume of the resulting solid.
Finding volume by shells,
π=π(-cos π + cos 0) = 2π cubic units
u =
du = 2x dx

(4)
Consider the curve for . Find the area of the surface that results when this curve is revolved around the x-axis.

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   pounds. How much work is done in moving the object 3 feet?

foot pounds