Question

Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about x = 4. y = 3x4, y = 0, x = 2

Answer 1

Answer: the volume V generated by rotating the regions 281.49

Explanation:

Given that;

the radius of the shell ⇒ (4 - x)

then the circumference of the shell is 2π( 3 - x ) i.e 2π × radius

y = 0 and x = 2

the height of the shell is 3x⁴

Now the volume of the solid obtained by rotating

the bounded region about x = 4 is;

V = ²∫₀ 2π ( 4 - x ) ( 3x⁴ ) dx

= ²∫₀ 2π ( 12x⁴ - 3x⁵ ) dx

= 2π [ 12x⁵/5 - 3x⁶/6 ]₀² {BY FTC -2]

= 2π [ (12×32)/5 - (3×64)/6 ] - ( 0-0) ]

= 2π [ 76.8 - 32]

= 2π [ 44.8 ]

= 281.49

Therefore the volume V generated by rotating the regions 281.49