The volume of the solid - [tex]215 \pi[/tex] unit.
The shell method is applied to determine the volume of a solid.
Mathematically, the volume of the solid is:
[tex]V = \int\limits^b_a A (x)dx[/tex]
where A(x) = 2πr (x) h(x)
The radius of a hole (O) = 1
The radius of a cone (F) =6
Height of cone (H) = 9
Formula:
2πam(da) = dV
Here,
m = [tex]\frac{Ha}{F}[/tex]
m = [tex]\frac{9a}{6}[/tex]
Substitute the given values in the formula,
2πam(da) = dV
2πa[tex]\frac{9a}{6}[/tex] = dV
2π[tex]\frac{9a^{2} }{6}[/tex] = dV
π[tex]\frac{9a^{2} }{3}[/tex] = dV
3[tex]a^{2}[/tex]π = dV
integrating the above equation,
[tex]\int\limits^F_O \pi 3a^{2} (da) = \int\limits dV[/tex]
Take , O=1 , F=6
[tex]\int\limits^6_1 \pi 3a^{2} (da) = \int\limits dV[/tex]
⇒[tex]\pi [6^{3} - 1^{3} ] = V[/tex]
⇒ [tex]\pi (216 -1) = V[/tex]
⇒ [tex]V = 215 \pi[/tex] unit
So, The volume of the solid - [tex]215 \pi[/tex] unit.
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