Page 240 - Calculus Workbook For Dummies

224 Part IV: Integration and Infinite Series

Q. Now tip the same glass vase up vertically. Wait! Another snag — similar to but unre-
This time find the volume of its glass with lated to the one in the last example. The
the cylindrical shells method. See the smaller shells, with right edges at x = 0
following figure. (Did you notice that the up to x = 3 2 , have heights that meas-
shape of the vase is now somewhat differ- 2
^
^
ent? Sorry about that.) ure from f xh up to g xh. But the larger
3 2
shells, with right edges at x = to
) 3√2, 9 ) 2
y 2 x = 3, have heights that measure from
f xh up to 9. So you’ve got to integrate
^
(3, 9)
the two batches of shells separately.
f(x) = x 2 Volume smaller shells = 2 π rhdx=
16 J N
g(x) = x 2 + 1
9 K O
K O
2
2
x
2 π x K 16 x + - 144 244 3 O dx
1
9
K 1 2 344 44 O
K K top ] gj bottom ] gj O
f `
x
O
g `
x
L P
Volume larger shells = 2 π rhdx=
2 π _ x i dx
2
x 9 -
(0, 1)
2. Add up all the shells by integrating.
x
(0, 0)
With the cylindrical shells method, you
integrate from the center to the outer
A. The volume is 45 π . edge.
2
Again, this is the same vase as in the 3 2 /2 16 3
2
2
2
x 9 -
disk/washer example, but this time # 2 π x c 9 x - x + 1m dx + # 2 π _ x i dx
it’s represented by different functions. 0 3 2 /2
3 2 /2 3
In a random act of kindness, I figured = 2 π # c 7 x + m 2 π # _ - x + 9i dx
3
3
x dx +
the new functions for you. 9
0 3 2 /2
1. Express the volume of your representa- = 2 π ; 7 x + 1 x E 3 2 /2 + 2 π - ; 1 x + 9 x E 3
4
2
2
4
tive shell. 36 2 0 4 2 3 2 /2
63
To figure the volume of a representative = 2 π c 16 + 9 m + 2 e 81 + 81 - - 81 + 81 mo
π -
c
2
16
4
4
4
shell, imagine taking the label off a can of
soup — it’s a rectangle, right? The area is = 45 2 π
$
base height and the base is the circum-
ference of the can. So the area is π2 rh. Amazing! This actually agrees (which, of
(r equals x and h depends on the given course, it should) with the result from
functions.) The thickness of the shell is the washer method. By the way, I got a
dx, so its volume is π2 rhdx. bit carried away with these example
problems. Your practice problems won’t
be this tough.