Section-7.3cx
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Transcript Section-7.3cx
DO NOW: Find the volume of the solid generated when the
region in the first quadrant bounded by the given curve and line
is revolved about the x-axis. y 2 x4 3x2 5
x2
(2,25)
Cross-section area:
A x r 2 x 3x 5
8
6
4
2
4 x 12 x 29 x 30 x 25
2
(0,5)
4
2
2
Volume:
f(x)
x
V 4 x8 12 x6 29 x 4 30 x 2 25 dx
2
0
2
4 9 12 7 29 5
3
x x x 10 x 25x
7
5
9
0
51574
315
The Washer Method
SECTION 7.3C
The region in the first quadrant enclosed by the y-axis and the
graphs of y = cos(x) and y = sin(x) is revolved about the x-axis
to form a solid. Find its volume.
Graph the region… and visualize the solid…
0,1
4, 2 2
Each cross section perpendicular to the
axis of revolution is a washer, a circular
region with a circular region cut from
its center:
R
r
Area of a washer:
R r
2
2
The region in the first quadrant enclosed by the y-axis and the
graphs of y = cos(x) and y = sin(x) is revolved about the x-axis
to form a solid. Find its volume.
0,1
4, 2 2
The outer and inner radii are the y
values of our two functions!!!
R cos x
r sin x
Cross section area:
A x cos x sin x
2
Volume:
V
4
0
4
0
2
cos2 x sin 2 x dx
4
1
cos 2xdx sin 2 x
2
2
0
units
cubed
Guided Practice
Find the volume of the solid generated by revolving the region
bounded by the given lines and curves about the x-axis.
1, 2
1,1
y 2x y x x 1
Cross section area:
A x
2x x 3 x
2
Volume:
1
V 3 x dx
2
0
1
x
3
3 0
3
2
2
Guided Practice
Find the volume of the solid generated by revolving the region
bounded by the given lines and curves about the x-axis.
y 4 x
1,3
2
y 2 x
Cross section area:
2
A x 4 x 2 x
2, 0
2
4
12 4 x 9 x x
2 2
Volume:
V 12 4 x 9 x x dx
2
2
4
1
2
x 108
2
3
12 x 2 x 3 x
5 1
5
5
Guided Practice
Find the volume of the solid generated by revolving the given
region about the y-axis.
The region bounded above by the curve y x and below
by the line y x .
Cross section area:
x y2
y y
A
y
y
y
1,1
x y
2
2
2 2
4
y
2
y
Volume: V y y dy
3 5 15
0
0
1
3
2
4
5
1
Guided Practice – Other Lines of Revolution!!!
Find the volume of the solid generated by revolving the region in
the first quadrant bounded above by the line y 2 , below by the
curve y 2sin x , 0 x 2 , and on the left by the y-axis,
about the line y 2 .
Cross section radius:
r 2 2sin x
Cross section area:
A x r 2 2sin x
2
r
2, 2
4 1 sin x
2
2
4 1 2sin x sin 2 x
Guided Practice – Other Lines of Revolution!!!
Find the volume of the solid generated by revolving the region in
the first quadrant bounded above by the line y 2 , below by the
curve y 2sin x , 0 x 2 , and on the left by the y-axis,
about the line y 2 .
Volume:
V
2
0
r
2, 2
4
4 1 2sin x sin x dx
2
0
2
1 1
1 2sin x cos 2 x dx
2 2
1
3
4 2sin x cos 2 x dx
0
2
2
2
1
3
4 x 2 cos x sin 2 x 3 8
4
2
0
2
Guided Practice – Other Lines of Revolution!!!
Find the volume of the solid generated by revolving the triangular
region bounded by the lines y = 2x, y = 0, and x = 1 about
(a) the line x = 1.
1, 2
1
Cross section radius: r 1
y
2
Cross section area:
1 1 y 1 y 2
A y 1 y
4
2
2
r
Volume:
1 2
V 1 y y dy
0
4
2
2
1 2 1 3
y y y
2
12 0 3
2
Guided Practice – Other Lines of Revolution!!!
Find the volume of the solid generated by revolving the triangular
region bounded by the lines y = 2x, y = 0, and x = 1 about
1
Washers!!! r 1 R 2
y
2
Cross section area:
(b) the line x = 2.
x2
2
1
2
A y 2 y 1
2
1 2
3 2y y
4
r
Volume:
R
y
y 8
2
V 3 2 y dy 3 y y
0
12 0 3
4
2
4
3
2
Guided Practice – Other Lines of Revolution!!!
Find the volume of the solid generated by revolving the region
2
bounded by the parabola y x and the line y 1 about
(a) the line y = 1.
Cross section:
r 1 x
A x 1 x
2 2
2
1 2x 2 x 4
Volume:
V 1 2 x x dx
1
2
4
1
2 3 1 5 16
x x x
5 1 15
3
1
Guided Practice – Other Lines of Revolution!!!
Find the volume of the solid generated by revolving the region
2
bounded by the parabola y x and the line y 1 about
(b) the line y = 2.
Washers:
r 1 R 2 x2
2
4
2 2
2
A x 2 x 1 3 4x x
Volume:
V 3 4 x x dx
1
2
4
1
4 3 1 5 56
3 x x x
3
5 1 15
1
Guided Practice – Other Lines of Revolution!!!
Find the volume of the solid generated by revolving the region
2
bounded by the parabola y x and the line y 1 about
(c) the line y = –1.
Washers:
r 1 x2 R 2
2
4
2
2 2
A x 2 1 x 3 2x x
Volume:
V 3 2 x x dx
1
2
4
1
2 3 1 5 64
3 x x x
3
5 1 15
1