Part II: The Answers
438
Answers
601–700
700.
625
6
π
To find the volume of a solid obtained by rotating about the y-axis a region that’s under
the curve y = f
 
(x) and above the x-axis from x = a to x = b with cylindrical shells, use
the integral
V
xf x dx
a
b
= ∫ 2π ( )
More generically, you can use the formula
d
V
x
a
b
= ∫
2π (
) (
)
shell radius shell height
To find the shell radius, let x be in the interval [a, b]; the shell radius is the distance
from x to the line of rotation. To find the shell height, let f
 
(x) be the function that
bounds the region above and let g(x) be the function that bounds the region below; the
shell height is given by f
 
(x) – g(x). Note that the limits of integration often correspond
to points of intersection.
Here’s the region being rotated about the y-axis:
Begin by isolating y in the second equation to get y = 5 – 2x. Then set the expressions
equal to each other and solve for x to find the x coordinates of the points of
intersection:
5 3
5 2
0 5
0
5
0 5
2
2
+
−
= −
=
−
=
−
=
x x
x
x x
x
x
x
(
)
,