Find the volume of the solid generated when the r egion in the first quadrant bounded by the given curve and line i s revolved about the x axis 05 225 f x x Crosssection area ID: 439662
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Slide1
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.
(0,5)
(2,25)
f
(x)
x
Cross-section area:
Volume:Slide2
Section 7.3cThe Washer MethodSlide3
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-axis
to form a solid. Find its volume.
Graph the region…
a
nd visualize the solid…
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:Slide4
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-axis
to form a solid. Find its volume.
The outer and inner radii are the
y
values of our two functions!!!
Cross section area:
Volume:
u
nits
cubedSlide5
Guided Practice
Find the volume of the solid generated by revolving the region
bounded by the given lines and curves about the
x
-axis.
Cross section area:
Volume:Slide6
Guided Practice
Find the volume of the solid generated by revolving the region
bounded by the given lines and curves about the
x
-axis.
Cross section area:
Volume:Slide7
Guided Practice
Find the volume of the solid generated by revolving the given
r
egion about the y-axis.
Cross section area:
Volume:
The region bounded above by the curve and below
b
y the line .Slide8
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 .
Cross section radius:
r
Cross section area:Slide9
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 .
r
Volume:Slide10
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.
r
Cross section radius:
Cross section area:
Volume:Slide11
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.
Washers!!!
R
r
Cross section area:
Volume:Slide12
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.
Cross section:
Volume:Slide13
Guided Practice – Other Lines of Revolution!!!Find the volume of the solid generated by revolving the regionbounded by the parabola and the line about
(b)
t
he line
y
= 2.
Washers:
Volume:Slide14
Guided Practice – Other Lines of Revolution!!!Find the volume of the solid generated by revolving the regionbounded by the parabola and the line about
(c)
t
he line
y
= –1.
Washers:
Volume: