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

Elisabeth Wiley

Elisabeth Wiley

Open question

2022-08-19

Finding Volume of Revolution with Multivariate Calculus
For 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 x
I'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 θ
where 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.

Answer & Explanation

Royce Morrison

Royce Morrison

Beginner2022-08-20Added 12 answers

Step 1
You 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 t
Step 2
And 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

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