1. 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.
(2,25)
Cross-section area:
(0,5)
Volume:
f(x) x

2. The Washer Method
Section 7.3c

3. 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…
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:

4. 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.
The outer and inner radii are the y
values of our two functions!!!
Cross section area:
Volume:
units
cubed

5. 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:

6. 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:

7. 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 and below
by the line
Cross section area:
Volume:

8. 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 , below by the
curve , , and on the left by the y-axis,
about the line
Cross section radius:
Cross section area: r

9. 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 , below by the
curve , , and on the left by the y-axis,
about the line
Volume: r

10. 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.
Cross section radius:
Cross section area: r
Volume:

11. 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
(b) the line x = 2.
Washers!!!
Cross section area: r R
Volume:

12. Guided Practice – Other Lines of Revolution!!!
Find the volume of the solid generated by revolving the region
bounded by the parabola and the line about
(a) the line y = 1.
Cross section:
Volume:

13. Guided Practice – Other Lines of Revolution!!!
Find the volume of the solid generated by revolving the region
bounded by the parabola and the line about
(b) the line y = 2.
Washers:
Volume:

14. Guided Practice – Other Lines of Revolution!!!
Find the volume of the solid generated by revolving the region
bounded by the parabola and the line about
(c) the line y = –1.
Washers:
Volume: