do now: find the volume of the solid generated when the region in the first quadrant bounded by the...
TRANSCRIPT
DO NOW: Find the volume of the solid generated when theregion in the first quadrant bounded by the given curve and lineis revolved about the x-axis. 4 22 3 5y x x 2x
(0,5)
(2,25)
f(x)
x
2A x rCross-section area:
24 22 3 5x x
8 6 4 24 12 29 30 25x x x x Volume:
2 8 6 4 2
04 12 29 30 25V x x x x dx
29 7 5 3
0
4 12 2910 25
9 7 5x x x x x
51574
315
SECTION 7.3C
The Washer Method
The region in the first quadrant enclosed by the y-axis and thegraphs of y = cos(x) and y = sin(x) is revolved about the x-axisto form a solid. Find its volume.
0,1
Graph the region… and visualize the solid…
4, 2 2Each cross section perpendicular to theaxis of revolution is a washer, a circularregion with a circular region cut fromits center:
R
r
Area of a washer:2 2R r
The region in the first quadrant enclosed by the y-axis and thegraphs of y = cos(x) and y = sin(x) is revolved about the x-axisto form a solid. Find its volume.
0,1 4, 2 2The outer and inner radii are the yvalues of our two functions!!!
cosR x sinr xCross section area:
2 2cos sinA x x x Volume:
4 2 2
0cos sinV x x dx
4
0cos 2xdx
2
4
0
1sin 22
x
unitscubed
Guided Practice
1,1
Find the volume of the solid generated by revolving the regionbounded by the given lines and curves about the x-axis.
2y x y x 1x Cross section area:
2 22A x x x Volume:
1 2
03V x dx
13
0
33
x
1,2
23 x
Guided Practice
2,0
Find the volume of the solid generated by revolving the regionbounded by the given lines and curves about the x-axis.
24y x 2y x Cross section area:
2 224 2A x x x
Volume:
2 2 4
112 4 9V x x x dx
252 3
1
12 2 35
xx x x
108
5
1,3
2 412 4 9x x x
Guided PracticeFind the volume of the solid generated by revolving the givenregion about the y-axis.
Cross section area:
22 2A y y y
Volume: 1 2 4
0V y y dy
13 5
03 5
y y
2
15
2 4y y
The region bounded above by the curve and belowby the line .
1,1
y xy x
2x y
x y
Guided Practice – Other Lines of Revolution!!!Find the volume of the solid generated by revolving the region inthe first quadrant bounded above by the line , below by thecurve , , and on the left by the y-axis,about the line .
2y 2siny x
2y 0 2x
2,2
Cross section radius:
22 2sin x 2A x r
2 2sinr x
r
Cross section area:
24 1 2sin sinx x
24 1 sin x
Guided Practice – Other Lines of Revolution!!!Find the volume of the solid generated by revolving the region inthe first quadrant bounded above by the line , below by thecurve , , and on the left by the y-axis,about the line .
2y 2siny x
2y 0 2x
2,2r
2 2
04 1 2sin sinV x x dx
Volume:
2
0
1 14 1 2sin cos 2
2 2x x dx
2
0
3 14 2sin cos 2
2 2x x dx
2
0
3 14 2cos sin 2
2 4x x x
3 8
Guided Practice – Other Lines of Revolution!!!Find the volume of the solid generated by revolving the triangularregion bounded by the lines y = 2x, y = 0, and x = 1 about
(a) the line x = 1.
1,2
r
Cross section radius:
2
112
A y y
112
r y Cross section area:
211
4y y
Volume:
2 2
0
11
4V y y dy
2
2 3
0
1 1
2 12y y y
2
3
Guided Practice – Other Lines of Revolution!!!Find the volume of the solid generated by revolving the triangularregion bounded by the lines y = 2x, y = 0, and x = 1 about
(b) the line x = 2.
2x 1r Washers!!!
122
R y
R
r
2
212 12
A y y
Cross section area:
213 2
4y y
42
03 2
4
yV y dy
Volume: 232
0
312
yy y
8
3
Guided Practice – Other Lines of Revolution!!!Find the volume of the solid generated by revolving the regionbounded by the parabola and the line about
(a) the line y = 1.
2y x 1y
21r x Cross section:
221A x x 2 41 2x x
Volume:
1 2 4
11 2V x x dx
13 5
1
2 1
3 5x x x
16
15
Guided Practice – Other Lines of Revolution!!!Find the volume of the solid generated by revolving the regionbounded by the parabola and the line about
2y x 1y (b) the line y = 2.
1r Washers:
22 22 1A x x 2 43 4x x Volume:
1 2 4
13 4V x x dx
13 5
1
4 13
3 5x x x
56
15
22R x
Guided Practice – Other Lines of Revolution!!!Find the volume of the solid generated by revolving the regionbounded by the parabola and the line about
2y x 1y (c) the line y = –1.
21r x Washers:
22 22 1A x x 2 43 2x x Volume:
1 2 4
13 2V x x dx
13 5
1
2 13
3 5x x x
64
15
2R