4.4 Let A be the solid torus in R3 obtained by revolving the disk (y - a)² + z² ≤ 6² in the yz-plane about the z-axis. (a) What are the intervals RC R2 and [c, d] C R such that A CRx [c, d]? Which axes correspond to R and which axis corresponds to [c, d]? (b) Recall that we define A₁ = {x E R² (x, t) € A}. : That is, A, is the cross section of A for each t = [c, d]. Find a formula for v(A₁) (the area of the cross section) for each value of t. (c) Use Cavalieri's principle to calculate the the volume of the torus, v(A). [Hint: The final answer should be v(A) = 2π²ab².]
4.4 Let A be the solid torus in R3 obtained by revolving the disk (y - a)² + z² ≤ 6² in the yz-plane about the z-axis. (a) What are the intervals RC R2 and [c, d] C R such that A CRx [c, d]? Which axes correspond to R and which axis corresponds to [c, d]? (b) Recall that we define A₁ = {x E R² (x, t) € A}. : That is, A, is the cross section of A for each t = [c, d]. Find a formula for v(A₁) (the area of the cross section) for each value of t. (c) Use Cavalieri's principle to calculate the the volume of the torus, v(A). [Hint: The final answer should be v(A) = 2π²ab².]
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![4.4 Let A be the solid torus in R³ obtained by revolving the disk (y - a)²+z² ≤ 6² in the
yz-plane about the z-axis.
(a) What are the intervals RC R2 and [c, d] CR such that A CRx [c, d]? Which
axes correspond to R and which axis corresponds to [c, d]?
(b) Recall that we define
At = {x E R² (x, t) € A}.
:
E
That is, At is the cross section of A for each t € [c, d]. Find a formula for v(A₁)
(the area of the cross section) for each value of t.
(c) Use Cavalieri's principle to calculate the the volume of the torus, v(A). [Hint:
The final answer should be v(A) = 2π²ab².]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa8d46346-fe9f-45aa-a3a0-df12b7cae379%2Fb02ef941-8165-400b-98fe-290e5083874d%2Fs9bt0ng_processed.jpeg&w=3840&q=75)
Transcribed Image Text:4.4 Let A be the solid torus in R³ obtained by revolving the disk (y - a)²+z² ≤ 6² in the
yz-plane about the z-axis.
(a) What are the intervals RC R2 and [c, d] CR such that A CRx [c, d]? Which
axes correspond to R and which axis corresponds to [c, d]?
(b) Recall that we define
At = {x E R² (x, t) € A}.
:
E
That is, At is the cross section of A for each t € [c, d]. Find a formula for v(A₁)
(the area of the cross section) for each value of t.
(c) Use Cavalieri's principle to calculate the the volume of the torus, v(A). [Hint:
The final answer should be v(A) = 2π²ab².]
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