Finding Volume of Revolution with Multivariate CalculusFor some function f(x) it is possible to rotate it along the x-axis and find the area using ∫ a b π ( f ( x ) ) 2 d xI'm curious how to do this with multivariate calculus. If I represent this in a parametric form, like g ( t ) = ( t , f ( t ) , 0 ) and then rotate using [ 1 0 0 0 c o s θ − s i n θ 0 s i n θ c o s θ ] The result is Ψ ( t , θ ) = ( t , f ( t ) ( c o s θ ) , f ( t ) ( s i n θ ) ) over a ≤ t ≤ b and 0 ≤ θ ≤ 2 π.I think to compute the volume, I need to use the double-integral V = ∬ ⟨ ∂ Ψ ∂ t ∂ Ψ ∂ θ ⟩ ∂ t ∂ θwhere I'm using ⟨⟩ for matrix determinant out of simplicity (and leaving out the bounds).I can find ∂ Ψ ∂ t = ( 1 , f ′ ( t ) ( c o s θ ) , f ′ ( t ) ( s i n θ ) ) and ∂ Ψ ∂ θ = ( 0 , f ( t ) ( − s i n θ ) , f ( t ) ( c o s θ ) ) .But the problem is that they would get put in a 3 × 2 matrix, so it doesn't make sense to take the determinant.

Question

Finding Volume of Revolution with Multivariate CalculusFor some function f(x) it is possible to rotate it along the x-axis and find the area using      ∫          a              b        π            (      f              (        x        )            )              2        d  xI&#039;m curious how to do this with multivariate calculus. If I represent this in a parametric form, like  g      (    t    )    =      (    t    ,    f          (      t      )        ,    0    )  and then rotate using       [                            1                          0                          0                                      0                          c          o          s                    θ                          −          s          i          n                    θ                                      0                          s          i          n                    θ                          c          o          s                    θ                      ]  The result is   Ψ      (    t    ,    θ    )    =      (    t    ,    f          (      t      )              (      c      o      s            θ      )        ,    f          (      t      )              (      s      i      n            θ      )        )   over   a  ≤  t  ≤  b and   0  ≤  θ  ≤  2  π.I think to compute the volume, I need to use the double-integral   V  =  ∬      ⟨                                                      ∂              Ψ                                      ∂              t                                                                          ∂              Ψ                                      ∂              θ                                            ⟩    ∂  t  ∂  θwhere I&#039;m using ⟨⟩ for matrix determinant out of simplicity (and leaving out the bounds).I can find             ∂      Ψ              ∂      t        =      (    1    ,          f      ′              (      t      )              (      c      o      s            θ      )        ,          f      ′              (      t      )              (      s      i      n            θ      )        )   and             ∂      Ψ              ∂      θ        =      (    0    ,    f          (      t      )              (      −      s      i      n            θ      )        ,    f          (      t      )              (      c      o      s            θ      )        )  .But the problem is that they would get put in a   3  ×  2 matrix, so it doesn&#039;t make sense to take the determinant.

Royce Morrison · Accepted Answer

Step 1You still need r because it can be anywhere inside the surface  Ψ  (  t  ,  r  ,      θ    )  =  (  t  ,  r  cos  ⁡  θ  ,  r  sin  ⁡  θ  )  ∭      ⟨                                                      ∂              Ψ                                      ∂              t                                                                          ∂              Ψ                                      ∂              r                                                                          ∂              Ψ                                      ∂              θ                                            ⟩      d  r    d  θ    d  tStep 2And then the limit of   r  =  f  (  t  )  ..      ∫    a    b        ∫    0          2      p      i            ∫    0          f      (      t      )        r    d  r    d  θ    d  t      ∫    a    b    π      f    2    (  t  )  d  t

Find the area using int_{a}^{b}pi(f(x))^{2}dx

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