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Examples and formulas for calculating the volume of solid figures using the parallel cross section and shell methods. Topics include disc method, cylinder volume, washer method, and cylindrical shell volume. Examples cover cones, spheres, and various regions of revolution.
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Section 6.3 Volume by the Shell Method
Test 3
Final Exam
Review for Test 3
Online Quizzes
Quiz 1 What is today?
a. Monday
b. Wednesday
c. Friday
d. None of these
Solid of Revolution About the x-Axis: Disk
Cylinder Volume: π[f (x∗ i )]^2 ∆xi Riemann Sum:
π[f (x∗ i )]^2 ∆xi
∫ (^) b
a
π[f (x)]^2 dx = lim ‖P ‖→ 0
π[f (x∗ i )]^2 ∆xi.
Example Example 1. Find the volume of the cone shown in the figure below.
Example Example 2. Find the volume of a sphere of radius r by revolving about the x-axis the region below the graph of
f (x) =
r^2 − x^2 , −r ≤ x ≤ r.
Solid of Revolution About the x-Axis: Washer
Cylinder Volume:∑ π([f (x∗ i )]^2 − [g(x∗ i )]^2 )∆xi [1ex] Riemann Sum: π([f (x∗ i )]^2 − [g(x∗ i )]^2 )∆xi [1ex]
∫ (^) b
a
π([f (x)]^2 − [g(x)]^2 ) dx = lim ‖P ‖→ 0
π([f (x∗ i )]^2 − [g(x∗ i )]^2 )∆xi
Solid of Revolution About the y-Axis: Washer Cylinder Volume:∑ π([F (y∗ i )]^2 − [G(y∗ i )]^2 )∆yi [1ex] Riemann Sum: π([F (y∗ i )]^2 − [G(y i∗ )]^2 )∆yi [1ex]
∫ (^) d
c
π([F (y)]^2 − [G(y)]^2 ) dy = lim ‖P ‖→ 0
π([F (y∗ i )]^2 − [G(y i∗ )]^2 )∆yi
Example Example 4. Find the volume of the solid generated by revolving the region between y = x^2 and y = 2x about the x-axis.
Example Example 5. Find the volume of the solid generated by revolving the region between y = x^2 and y = 2x about the y-axis.
Volume of a Cylindrical Shell
Volume of a Cylindrical Shell
Solid of Revolution About the y-Axis: Shell
∫ (^) b
a
2 π x [f (x) − g(x)] dx = lim ‖P ‖→ 0
2 π x∗ i [f (x∗ i ) − g(x∗ i )]∆xi.
The integrand 2π x [f (x) − g(x)] is the lateral area of the cylinder.
Solid of Revolution About the x-Axis: Shell
∫ (^) d
c
2 π y [F (y) − G(y)] dy = lim ‖P ‖→ 0
2 π y∗ i [F (y∗ i ) − G(y∗ i )]∆yi.
The integrand 2π y [F (y) − G(y)] is the lateral area of the cylinder.
Example
Example 8. Find the volume of the solid generated by revolving about the x-axis the region between y = x^2 and y = 2x.
Example Example 9. A round hole of radius r is drilled through the center of a hemisphere of radius a. Find the volume of the potion of the hemisphere that remains.
Example Example 10. The region Ω between y =
x and y = x^2 , 0 ≤ x ≤ 1, is revolved about the line x = −2. Find the volume of the solid that is generated.
