Answer :
The volume of the solid obtained by rotating the region bounded by the given curves about the X-axis is 340978 vu.
Given curve is,
y = [tex]x^{3}[/tex] , y = 8, x = 0 about x = 7
we know the formula for cylindrical shell is
V = πR²h
where R² = f(x)₂-f(x)₁ ,
h = dx
here f(x)₂ = 8 and f(x)₁ = x³
⇒ V = π(8 - x³)²h
V = π(64 - 16x³+x⁶)dx
integrating on both sides,
[tex]\int dV[/tex] =π [tex]\int ({64 - 16 x^{2} + x^{6} }) \, dx[/tex]
= [tex]64 \int dx - 16 \int {x^{3} } \, dx + \int\ {x^{6} } \, dx[/tex]
V = π(64x - 4x⁴+[tex]\frac{x^{7} }{7}[/tex])
V = π(64 × 7 - 4(7)⁴ + [tex]\frac{7^{7} }{7}[/tex])
= π(448 - 9604 + 117649 )
= [tex]\frac{22}{7}[/tex](108493)
V = 340978 vu
Therefore volume of the cylindrical shells to the region y = x³, y = 8, x = 0 about x = 7 is 340978 vu.
Cylindrical shells are essential structural elements in offshore structures, submarines, and airspace crafts. They are often subjected to combined compressive stress and external pressure, and therefore must be designed to meet strength requirements.
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