Consider the infinite region bounded by the curves y=1/x, x=1, and the x-axis. Show that the solid obtained by rotating this reg
ion about the x-axis has finite volume, but infinite surface area. Why? "This question is my professor make up it and it's worth it for 2 credit, so please help me."
1 answer:
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Take derivitive
note
the derivitive of sec(x)=sec(x)tan(x)
so
remember the quotient rule
the derivitive of
so
the derivitive of
so now evaluate when t=pi
we get
sec(pi)=-1
tan(pi)=0
we get
slope=1/pi
use slope point form
for
slope=m and point is (x1,y1)
equation is
y-y1=m(x-x1)
slope is 1/pi
point is (pi,1/pi)
y-1/π=1/π(x-π)
times both sides by π
πy-1=x-π
πy=x-π+1
y=(1/π)x-1+(1/π)
or, alternately
-(1/π)x+y=(1/π)-1
x-πy=π-1
note
the derivitive of sec(x)=sec(x)tan(x)
so
remember the quotient rule
the derivitive of
so
the derivitive of
so now evaluate when t=pi
we get
sec(pi)=-1
tan(pi)=0
we get
slope=1/pi
use slope point form
for
slope=m and point is (x1,y1)
equation is
y-y1=m(x-x1)
slope is 1/pi
point is (pi,1/pi)
y-1/π=1/π(x-π)
times both sides by π
πy-1=x-π
πy=x-π+1
y=(1/π)x-1+(1/π)
or, alternately
-(1/π)x+y=(1/π)-1
x-πy=π-1