March 20, 2002
(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
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.
(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?