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In problems 11-12, use shells to find the volume of the solid obtained by rotating the given region about the specified line. Sketch the region, the area to be rotated, and a typical rectangle. 12. 11. y = 6 x+2 y = = x + ² 3 x axis y = √3x - 11 y=x-3 y axis

In problems 11-12, use shells to find the volume of the solid obtained by rotating the given region about the specified line. Sketch the region, the area to be rotated, and a typical rectangle. 12. 11. y = 6 x+2 y = = x + ² 3 x axis y = √3x - 11 y=x-3 y axis

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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In problems 11 – 12, use shells to find the volume of the solid obtained by rotating the given region about the specified line. Sketch the region, the area to be rotated, and a typical rectangle. 11. 6 x+2 ==x+² 3 y = y = x axis 12. y = √√3x - 11 y=x-3 y axis
Transcribed Image Text:In problems 11 – 12, use shells to find the volume of the solid obtained by rotating the given region about the specified line. Sketch the region, the area to be rotated, and a typical rectangle. 11. 6 x+2 ==x+² 3 y = y = x axis 12. y = √√3x - 11 y=x-3 y axis
Expert Solution
Step 1

The equation of the region given is as follows:

y=6x+2y=-13x+73

The region is rotated about the x-axis.

The shell method to find the volume of the region rotated about the x-axis is as follows:

V=2π∫y=abryhydy..................................................................................(1)

The radius is the distance from y to the x-axis, so ry=y. 

To find the height we will first find the equations in terms of y.

x=6y-2x=7-3y

Hence, the height of the region is as follows:

hy=7-3y-6y-2       =7y-3y2-6+2yy       =9y-3y2-6y

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