MATH 37-THIRD LONG EXAMS

A. DIRECTION: Sketch the graph and show how you slice the rectangular element. Set up the definite integral only. Do not evaluate.

1. Let R be the region bounded by the graph of y = 1 + x and y = 1 - x.
   a. Find the are of R.                b. Find the centroid of R.
2. Let R be the region bounded by y = ln x, x = e and the x-axis.
   a. Find the volume of solid or revolution when R is revolved about the
       a.1 x - axis                            a.2 y = -1
       a.3 x = e                                a.4 x = -1
   b. Find the centroid of solid revolution of a.3 and a.4.
3. Find the area of the surface generated when the given arc is revolved about the indicated axis.
   a. y = ln sec x from x = 0 to x = p/3 about the x - axis.
   b. 8x = y⁴ = 2y² from y = 1 to y = 2

B. Evaluate the definite integral.

4. The measure of the linear density at a point of a rod varies directly as the third power of the measure of the distance of the point from one end. The lengtj of the rod is 4 ft and the linear density is 2 slugs/ft at the center. Find the center of mass.