7.3 Volumes of Revolution: the Shell Method - Battaly

[Pages:16]7.3 Volumes of Revolution: the Shell Method

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Class Notes: Prof. G. Battaly, Westchester Community College, NY Homework Part 1 Homework Part 2

Title: Homework (1 of 16)

7.3 Volumes of Revolution: the Shell Method

Consider: y = x3 3x + 3, x = 0, y = 0, x = 2

Easy to revolve about xaxis: Use disk method BUT, what about revolving about y axis?

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Class Notes: Prof. G. Battaly, Westchester Community College, NY Homework Part 1 Homework Part 2

Title: Introduction (2 of 16)

7.3 Volumes of Revolution: the Shell Method

Consider: y = x3 3x + 3, x = 0, y = 0, x = 2

What about revolving about y axis?

Reference rectangle for disk method is not consistent and does not have an easy algebraic representation.

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Class Notes: Prof. G. Battaly, Westchester Community College, NY Homework Part 1 Homework Part 2

Title: problems w. disk method (3 of 16)

7.3 Volumes of Revolution: the Shell Method Consider: y = x3 3x + 3, x = 0, y = 0, x = 2

12 3

Could divide into 3 regions. Then add the volumes.

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Class Notes: Prof. G. Battaly, Westchester Community College, NY Homework Part 1 Homework Part 2

Title: how to set up disk method (4 of 16)

7.3 Volumes of Revolution: the Shell Method Consider: y = x3 3x + 3, x = 0, y = 0, x = 2

For Region 1:

12 3

We have x, but we need an integrand to

match the dy.

Could divide into 3 regions.

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Class Notes: Prof. G. Battaly, Westchester Community College, NY Homework Part 1 Homework Part 2

Title: match variable in integrand (5 of 16)

7.3 Volumes of Revolution: the Shell Method Consider: y = x3 3x + 3, x = 0, y = 0, x = 2

12 3

For dy, we need to solve the cubic function for x in terms of y to express the integrand algebraically

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Class Notes: Prof. G. Battaly, Westchester Community College, NY Homework Part 1 Homework Part 2

Title: if y=f(x), then x=? (6 of 16)

7.3 Volumes of Revolution: the Shell Method

Use alternate method: the Shell Method .

Start with reference rectangle, but this time the Reference Rectangle is parallel to the axis of revolution.

Calculus Home Page

Class Notes: Prof. G. Battaly, Westchester Community College, NY Homework Part 1 Homework Part 2

Title: alternate approach (7 of 16)

7.3 Volumes of Revolution: the Shell Method

Use alternate method: the Shell Method .

Startwith reference rectangle, but this time the Reference Rectangle is parallel to the axis of revolution.

Calculus Home Page

Class Notes: Prof. G. Battaly, Westchester Community College, NY Homework Part 1 Homework Part 2

Title: Shell animation (8 of 16)

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