Page 253 - Calculus Workbook For Dummies

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

2. Express the volume of your representative slice.
2
2
2
2
2
=
Volume washer = π _ R - r i dx π c x - x m dx π _ x - x i dx
=
3. Add up the infinite number of infinitely thin washers from 0 to 1 by integrating.
1 1 π
π
3
2
2
=
Volume solid = # _ x - x i dx π ; 1 x - 1 x E = π c 1 - 1 m =
2 3 2 3 6
0 0
Note that the infinite number of washers you just added contain an infinite number of
holes — way more than the number of holes it takes to fill the Albert Hall. For extra credit:
What “holes” were the Beatles referring to? Hint: Remember the charioteer Glutius from
Chapter 8?
2048 π
2
i Same as problem 8, but with f x = x and g x = 4 x. The volume is cubic units.
^ h
^ h
15
1. Sketch the solid and a representative slice. See the following figure.
f(x) = x 2
y g(x) = 4x
(4, 16)
(0, 0)
x
(4, 0)
2. Determine where the functions intersect.
2
x = 4 x
x = 4 and thus y = 16
3. Express the volume of a representative washer.
2 2
4
2
2
2
2
=
=
Volume washer = π _ R - r i dx π ^ a 4 h x i k dx π 16 x - x i dx
x - _
_
4. Add up the washers from 0 to 4 by integrating.
4 4 π
π
3
5
4
2
=
Volume solid = # _ 16 x - x i dx π ; 16 x - 1 x E = π c 1024 - 1024 m = 2048
3 5 3 5 15
0 0
*j Use the disk method to derive the formula for the volume of a cone. The formula is 1 π r h.
2
3
1. Find the function that revolves about the x-axis to generate the cone.
r
The function is the line that goes through (0, 0) and (h, r). Its slope is h and thus its equation
r
is f x = x.
^ h
h
^
2. Express the volume of a representative disk. The radius of your representative disk is f xh
and its thickness is dx. Its volume is therefore
2
2 r
π
=
^
m
V disk = ` f xhj dx π c x dx
h
3. Add up the disks from x = 0 to x = h by integrating. Don’t forget that r and h are simple
constants.
h 2 r π 2 h r π 2 h r π 2
V cone = # π c r x dx = 2 # x dx = 2 ; 1 x E = 2 $ 1 h = 1 π r h
3
3
2
2
m
h h h 3 h 3 3
0 0 0