Volume Match Lab
Match each Graph of Bounded Region it to its Resulting Solid Figure and
Equation(s), Boundaries & axis of Rotation. Then, write the integral needed to
find the volume of the resulting solid. Find the volume, where possible.
Graph of
Bounded
Region
Resulting
Solid
Figure
Equation(s),
Boundaries,
& axis of
Rotation
Integral Expression for
Finding Volume
Volume
(where
possible)
1
2
3
4
5
6
Volume Match Lab
2007
Created by Anne Chay—Lawrence, MA
The region described below is to be rotated
about the y-axis.
The region described below is to be rotated
about the x-axis.
It is bounded by:
It is bounded by:
2
y = x + 1,
y = 0
y =
sinx ,
x = 0 and
x = 1.
x = 0
y = 0
and
x = ! .
Equations, Boundaries, Rotation 1
Equations, Boundaries, Rotation 2
The region described below is to be rotated
about the y = 1.
The region described below is to be rotated
about the x-axis.
It is bounded by:
It is bounded by:
2
y=2–x
and
y=1
y = R (x) ,
y = r (x)
x = a
x = b.
and
.
Equations, Boundaries, Rotation 3
Equations, Boundaries, Rotation 4
The region described below is to be rotated
about the x-axis.
The region described below is to be rotated
about the x-axis.
It is bounded by:
It is bounded by:
y =
2
25 – x
x = -4
and
,
y = 3
x = 4.
Equations, Boundaries, Rotation 5
y = R (x) ,
x = a
and
y = 0
x = b.
Equations, Boundaries, Rotation 6
Volume Match Lab
Answer Sheet
Graph of
Bounded
Region
Resulting
Solid
Figure
Equation(s),
Boundaries,
& axis of
Rotation
Integral Expression for
Finding Volume
!
1
2
3
4
5
6
3
2
1
4
2
5
4
3
$
! # (
"0
4
' '
& &
% %
$
!#
"-4
5
(
= 2!
! 6.283
2
) dx
2 *)
– 3
(
dx
= 256 !
3
!
268.083
b
2
2
$
! # ( R(x)) – r(x) ) dx
"a
1
2
2
$
! # (2 – x – 1) dx
"-1
= 16 !
15
!
3.351
b
2
$
! # ( R(x) ) dx
"a
6
1
*2
2 )
25 – x
Washers:
6
sinx
Volume
(where
possible)
2
2
2
$
$
! # (1 ) dy – ! # ( y – 1 ) dy
"0
"1
Shells:
1
2
$
2 ! # x ( x + 1 ) dx
"0
= 2! – 1!
2
!
4.712