Calculate fxyzdS For x2y2250z8fxyzez fxyzdSSolutionsolution

Calculate f(x,y,z)dS For x^2+y^2=25,0z8;f(x,y,z)=e^(z) f(x,y,z)dS

Solution

solution

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we have to parameterize over a cylinder.
We split the cylinder into three parts, the top, the bottom and the lateral surface

On top we have z=8, so

A₁= e^-8 dS=e^-8dS=e^-8 *area circle of radius 5=e^-8*25

On bottom, similarly since z=0, A₂=25

For the lateral surface x²+y²=25, we cannot write z as function of x and y, i.e. z=f(x,y)

So we cannot use the most common method.

However, we can use the second method by parameterizing the surface on two parameters (see the second green formula at http://tutorial.math.lamar.edu/Classes/CalcIII/SurfaceIntegrals.aspx )

the cylinder can be parametrized

r=(x,y,z)=(5cos(t),5sin(t),z)

t from 0 to 2, z from 0 to 8

dr/dt x dr/dz=

   i               j           k

   dx/dt        dy/dt    dz/dt

   dx/dz     dy/dz    dz/dz

=

      i             j k

-5sin(t)            5cos(t)    0

   0                      0          1

=(5cos(t),5sin(t),0)

Then dS=|dr/dt x dr/dz|=5

Then

A₃=5e^(-z) dz dt=10(-e^(-z))|0..8

=10(1-e^-8)

Then we add the integrals

A=A₁+A₂ +A₃=35+15e^-8