BC 2-3
Name ___________________________
Show enough work that I can reproduce your results. No calculators.
Skills:
AP 2010 #4
R
1.
IMSA
Let R be the region in the first quadrant bounded by the graph of y  2 x , the horizontal line
y  6 , and the y-axis, as shown in the figure above.
(a) Find the area of R.
(b) Write, but do not evaluate, an integral expression that gives the volume of the solid generated
when R is rotated about the horizontal line y = 7.
(c) Region R is the base of a solid. For each y, where 0  y  6 , the cross section of the solid
taken perpendicular to the y-axis is a rectangle whose height is 3 times the length of its base
in region R. Write, but do not evaluate, an integral expression that gives the volume of the
solid.
F12
2. Set up an integral(s) that give the volume of the solid obtained by rotating the region bounded by
y  2 x  1 and y  x  1 about the line x  6 . (You may want to sketch the region).
3.
Set up an integral, but DO NOT
integrate, to find the length of the
curve shown at the right, which is
given by the equations
t
x  1  3cos   and
2
1
y  sin  2t   cos  5t  for
2
0    4 .
1.5
1.0
0.5
2
1
1
2
3
4
0.5
1.0
1.5
IMSA
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4.
Let R be the region R bounded
by the graphs of y  ln  2 x  3 ,
y  ln  x  1 , and y = 0.
Set up the integral(s), but DO
NOT integrate, to find the
volume of the solid formed by
revolving region R around each
axis using the indicated method.
IMSA
a.
x-axis (disks/washers)
b.
y-axis (any method)
c.
y = –1 (Shells)
d.
x = 9 (any method)
1.5
1.25
1
(4, ln(5))
y  ln  x  1
y  ln  2 x  3
0.75
0.5
0.25
1
2
3
4
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Concepts:
5. Let R be the region bounded by the graphs of y  ln  x  , x  e, and the x-axis, as pictured below. Find
the volume of the solid generated when R is rotated about the x-axis.
R
IMSA
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6. Let R be the region in Quadrant I bounded by the graphs of y  x 1  x 2 and the x-axis. Find the
volume of the solid generated when R is rotated about the y-axis.
IMSA
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7. Find the volume of the solid generated by revolving the region in the first quadrant bounded by
the hypocycloid x  cos3  , y  sin3  , the x-axis, and the y-axis (pictured below) about the x-axis.
R
IMSA
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