Page 239 - Calculus Workbook For Dummies

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Chapter 12: Who Needs Freud? Using the Integral to Solve Your Problems

Q. Using the disk/washer method, what’s the The large circle has an area of Rπ 2 , and
2
volume of the glass that makes up the vase the hole an area of rπ . So a washer’s
2
shown in the following figure? cross-sectional area is Rπ 2 - r π , or
2
2
π _ R - r i. It’s thickness is dx, so its
2
2
g(x) = .75√x − 1 volume is π _ R - r i dx.
y
(9, 3)
f(x) = √x Back to our problem. Big R in the vase
problem is x and little r is .75 x - 1,
so the volume of a representative washer
2 2
(9, 0) is π ` c x - `j .75 x - 1j m dx
x
(0, 0) (1, 0)
4. Add up the washers by integrating from
0 to 9.
But wait; did you notice the slight snag in
this problem? The “washers” from x = 0
A. The volume is 45 π . to x = 1 have no holes so there’s no little-
2 r circle to subtract from the big-R circle.
First, here’s how the vase is “created.” A washer without a hole is called a disk,
The light gray shaded area shown in the but you treat it the same as a washer
figure lies between x and .75 x - 1 except you don’t subtract a hole.
from x = 0 to x = 9. The three-dimensional
vase shape is generated by revolving the 5. Add up the disks from 0 to 1 and
shaded area about the x-axis. the washers from 1 to 9 for the total
volume.
1. Sketch the 3-D shape (already done for
1 9
2
you). Volume vase = # π x dx + # π d x - a .75 x - 1k n dx
2
2
0 1
2. Indicate a representative slice (see the 1 9 9
π
dark gray shaded area in the figure). = π # x dx + # d x - 16 ^ x - 1hn dx
0 1
1 9 9
3. Express the volume of the representa- = π # x dx + # x dx - 9 16 π #^ x - 1h dx
π
tive slice. 0 1 1
9 9 π 9
π #
A representative slice in a washer prob- = x dx - 16 #^ x - 1h dx
0 1
lem looks like — can you guess? — a π 9 9 π 9
2
= x F - ^ x - 1h F
2
washer. See the following figure. 2 32
0 1
81 π
= - 18 π
2
45 π
=
2
R
r