Math 1501

Don't Panic.
Open Book and Notes. You have 40 minutes. Carefuly explain your proceedures and answers
All answers should be given exactly (no decimals) if possible.

Problem 1 (10 points)

Consider the region definted as below by thex and y axes, the line y = 2x - 2,
and the quarter circle centered at x=0, y=4, with radius 3.

What is the vulume of the region obtained by revolving this regoin about
the y axis? No calculus need be used. This is a truncated cone and an
hemisphere.

What is the x coordinate of thecentroid of the (planar) region defined above? No integrals are necessary.

Calculations

Solution Calculations.

Probem 2 (10 points)

By the method of SHELLS, set up the integral(s) needed to find the volume of the revolved region in problem 1.

Answer

The top cicle is

The stright line is 2x - 2

Problem 3 (10 Points)

Find the centroid of the planar region bounded by
y =+ 5 = 0, x = 4, and y = 3. Don't evaluate the integrals, just set them up.

Calculations

pic = Show[Plot[{f[x],g[x]} ,{x,0,4.3},
   PlotRange ->{-1,30},
   DisplayFunction -> Identity],
Graphics[
       {Line[{{4,0},{4,4^2 + 6}}]}],
   DisplayFunction -> $DisplayFunction]