Notes on vectors and integrals, Lecture notes of Calculus

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Section 6.3 - Volume by Slicing

Calculus II Week 2 From page 426

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General Slicing

Let A be a solid that lies between @ = B and @ = C. If the cross-sectional area of A in the plane D', through @ and perpendicular to the @-axis, is E(@), where E is a continuous function, then the volume of A is: F = lim (→*

H

+,# ( E @+ ∗ ∆@ = 9 . / E @ <@ Recall: lim (→*

H

0 ,# ( J @ 0 ∗ ∆@ = 9 . / J @ <@ Calculus II Week 2

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Disk Method (Example)

Find the volume of the solid obtained by rotating the region bounded by = = @ " , = = 8 and @ = 0 about the = - axis Calculus II Week 2

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Solids of Revolution (Washer Method)

If the region we revolve to generate a solid does not border on (or cross) the axis of revolution, the solid has a hole in it. The cross-sections perpendicular

to the axis of revolution are^ washers^ instead of disks.

Calculus II Week 2

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Washer Method (Example)

The region R enclosed by the curves = = @ and = = @ ! is rotated about the @–axis. Find the volume of the resulting solid. Calculus II Week 2