Written HW # 1: Areas and Volumes!!!
MAC 2282/2312
Score:
/20
I. Before We Get Started: Formulas
1. Area: It is a two-dimensional quantity. To find the area in between two curves f (x) and g(x),
if the integration bounds are not given, we first find the intersections by
. Say that the intersections are a and b and that
f (x) ≥ g(x) for all a ≤ x ≤ b, then the area in between f (x) and g(x) over [a, b] is given by:
A=
Note: You can always use absolute value to make sue that the result is positive.
2. Volume:
a) By Slicing: Washer
(The disk method is a special case of the washer method with inner radius equal to 0.)
Rotation about
x − axis or
y = b (a horizontal line)
Integration
variable
answer x or y
Function in
the form
answer f (x) or f (y)
Draw a general
cross section
Cross section
area
Volume
integral
1
y − axis or
x = a (a vertical line)
2
b) By Cylindrical Shells
Rotation about
x − axis or
y = b (a horizontal line)
y − axis or
x = a (a vertical line)
Integration
variable
answer x or y
Height in
terms of
answer f (x) or f (y)
Radius in
terms of
answer of x or y
Draw a general
cylinder
Volume
integral
II. Problems
1. Find the area of the region bounded by the curves x = |y| and x = 2 − y 2 .
3
2. Let R be the region bounded by y = x2 and y = 4x–x2 .
(a) Sketch the region R.
(b) Find the area of R.
(c) Find the volume obtained by rotating the region R about the x-axis.
(d) Find the volume obtained by rotating the region R about the y-axis.
4
3. Find the volume of the solid generated by rotating the region bounded by y = x3 , y = 0, and
x = 2 about:
(a) the x-axis
(b) the y-axis
(c) the line x = 4
(d) the line y = 8
5
4. Find the volume of the solid obtained by rotating the region bounded by the x-axis and the
graph of y = 1 − x2 about the line y = –3.
5. (+2 Bonus:) A spherical ball bearing with radius 10 cm is going to be used to build a wheel.
The manufacturer needs to bore a cylindrical hole with radius 2 cm through the ball bearing
so that it can be mounted to the legs of a workbench. Use the shell method to calculate the
volume of material that must be removed from the ball bearing. Show all work and include
units in your answer.
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Solutions part II:
1.
7
3
2. (a) Area between the curves
y
4
4x − x2
3
2
x2
1
x
1
(b)
8
3
(c)
32
π
3
(d)
16
π
3
3. (a)
4.
128
π
7
(b)
64
π
5
(c)
96
π
5
(d)
320
π
7
136
π
15
5. 248.8 cm3
2