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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