Answered: se the method of cylindrical shells to…

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter8: Further Techniques And Applications Of Integration

Section8.3: Volume And Average Value

Problem 13E

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Question

Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the y-axis.

x= -3y²+15y-18,

x=0

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Step 1

$G i v e n x = - 3 y^{2} + 15 y - 18, x = 0 a b o u t y - a x i s s - 3 y^{2} + 15 y - 18 = 0 [∴ x = 0] \Rightarrow 3 y^{2} - 15 y + 18 = 0 T h i s e q u a t i o n q y a d r a t i c e q u a t i o n b y s o l v i n g \Rightarrow 3 (y^{2} - 5 y + 6) = 0 \Rightarrow y^{2} - 5 y + 6 = 0 y^{2} - 3 y - 2 y + 6 = 0 \Rightarrow y (y - 3) - 2 (y - 3) = 0 \Rightarrow (y - 3) (y - 2) = 0 \Rightarrow y = 3 o r y = 2 U \sin g c y l i n d e r m e t h o d, w e w i l l i n t e g r a t e f r o m 2 t o 3 w e h a v e V = 2 π \int_{a}^{b} rh dy Here a = 2, b = 3 r \to radius of each cylinder h \to height of cylinder$

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