Volume by Shell and Washer Methods

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SUMMARY

The volume generated by rotating the region defined by the equations x = 4y, y = 0, x = 0, and x = 8 about the x-axis can be calculated using both the Washer and Shell methods. The Washer method yields a volume of V = 32π/3, while the Shell method, after correcting for the upper limit, also results in V = 32π/3. This confirms that both methods produce the same volume when applied correctly, highlighting the importance of accurate limits in integration.

PREREQUISITES
  • Understanding of integral calculus, specifically volume calculations using the Washer and Shell methods.
  • Familiarity with the equations of curves and their graphical representations.
  • Knowledge of definite integrals and their applications in calculating areas and volumes.
  • Ability to manipulate and simplify algebraic expressions within integrals.
NEXT STEPS
  • Review the Washer method for volume calculations in different scenarios.
  • Study the Shell method in depth, focusing on its applications and limitations.
  • Practice solving volume problems involving rotation about different axes.
  • Explore common mistakes in integration and how to avoid them in calculus problems.
USEFUL FOR

Students studying calculus, particularly those focusing on volume calculations through rotation methods, as well as educators seeking to clarify these concepts for their students.

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Homework Statement



Find the volume generated by rotating the given region about the given line using the Shell method and the Washer method.

x = 4y [y = x/4], y = 0, x = 0, x = 8 about x



Homework Equations



Washer method (about x):
V = pi \int^b_a ((Rtop2(x) - rbottom2(x))dx

Shell method (about x):
V = 2pi \int^d_c (y[f(y)-g(y)])dy

The Attempt at a Solution



I'm not sure why I can't reconcile these two answers. I'm having some similar problems with more of these exercises but if someone can help me see where I'm going wrong I'm sure I can rework them successfully.

Washer:
V = pi \int^8_0 ((x/4)2)dx = 32pi/3

Shell:
V = 2pi \int^2_0 (y(4y))dy = 64pi/3
 

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I forgot about x=8. The shell method should be V = 2pi \int^2_0 (y(8-(4y)))dy = 32pi/3.

:facepalm:
 
Double-post :/

:facepalm:
 
Last edited:

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