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Calculus  II                                                 Test #2                                      March 18, 2005

Name____________________           R.  Hammack                                     Score ______
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(1) Find the area of the region contained between the graphs of y=+1  and y=x,  and between x=-1 and x=2.

(2) Find the area of the region contained between the curves  y = and  y =.

These curves intersect at points with x coordinates that satisfy =

Thus, the curves intersect at x = 1 and  x = -1.

(3) Consider the region contained between the graphs of y=,  y=0, x=0, and x=π/3,
This region is revolved around the x-axis. Find the volume of the resulting solid.

By slicing:

(4) Consider the region contained between the graphs of y=-2+x  and  y=0.
This region is revolved around the y-axis. Find the volume of the resulting solid.

Note  y = -2+x = x(-2x+1) = , so the x intercepts are 0 and 1.
Drawing a rough sketch of the graph, we see that the region lies between 0 and 1 on the x-axis

Volume by shells:
2π x(-2+x)dx = 2π(-2+)dx = 2 = 2π( -+) = 2π( -+) = Cubic Units

(5)
Find the exact arc length of the curve  y = f(x) = dt  between x=1 and x=2.

Note: f'(x)=.  Now using the arc length formula, we get
dx = dx = dx = dx=(1+x)dx = = (2+)-(1+) = 2+2-1-1/2 = 5/2  Units.

(6)
Consider the graph of y=+1 between x=1 and x=3.  This graph is revolved around the x-axis. Find the area of the resulting surface.
Using the arc length formula, we get
2π(+1)dx = 2π(+1)dx = π(+1)dx = π = π((+3)-(+1)) = π(+3--1) = 4π   Square Units.

(7) A variable force pushes an object 10 feet along a straight line in such a way that when the object is x feet from its starting point, the force on the object is  2- pounds. How much work is done in moving the object 10 feet?

Work = ( 2-)dx = = (20+)-(0+) = 10 Foot pounds