1) What is the arc length of C? 2) The surface of revolution on the x axis?
c: x= cost + ln(csct-cot t), y=sint (#pi/6# #<=# t #<=# #pi/3# )
c: x= cost + ln(csct-cot t), y=sint (
1 Answer
Arc length:
Surface area of revolution:
Bonus...
Volume of solid of revolution:
Explanation:
Arc length
The arc length of a curve given parametrically is
Derivation here: http://tutorial.math.lamar.edu/Classes/CalcII/ParaArcLength.aspx
In our case,
Use the quotient rule and the definitions of
The expression for
So arc length
Multiply out within the square root:
Recall the identity
Recall also the similar identity
To integrate
Note that evaluation of integrals of a logarithm is only valid when both limits are on the same half of the logarithm curve, i.e. if the point that the log blows up to minus infinity at is not between the two points. Happily, this is the case here; both limits are the same side of the vertical asymptote at
Recall that by definition
This is the simplest form that this answer can take. It's nice, isn't it? We started out with such an obscure and messy function and gradually simplified it down to this expression. Someone somewhere invested a lot of effort in designing this question!
Surface of revolution
I'm going to assume that you are asking for the calculation of the surface area of the surface of revolution here, which is of a piece with the first part of the question. We already know from the question how to specify the surface.
The surface area for a rotation about the
Derivation here: http://tutorial.math.lamar.edu/Classes/CalcII/ParaSurfaceArea.aspx
Note that this is a pretty similar formula to the arc length formula we evaluated above; we already know that
Cancel factor of
Volume of revolution
Extra bonus answer! You didn't ask this, but it's the other linked quantity, and often is asked in such questions - the volume enclosed by the surface of revolution, the solid of revolution. It probably isn't asked for here as the question is designed to have you perform the elegant reduction
The volume of a solid of revolution about the
So
Recall again the identity
Multiply out and cancel
Notice that this integrand is of the form produced by a chain rule differentiation: