What is the surface area produced by rotating #f(x)=abs(1-x), x in [0,3]# around the x-axis?

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Answer 1 · 2017-01-23T19:09:24

# 5 sqrt(2) \ pi#

If we look at the graph #y=abs(1-x)# we get:
graph{|1-x| [-10, 10, -5, 5]}

so rotating about )x will produce two cones:

One smaller cone of height #1#, and one larger cone of height #3#,

The surface area of a cone of radius #r# is given by #pi rl# where #l# is the length of the slope:

For the smaller cone:

# l^2=1^2+1^2 => l=sqrt(2) #

For the larger cone:

# l^2=2^2+2^2 => l=sqrt(8) =2sqrt(2)#

So total surface area is:

# SA =(pi)(1)(sqrt(2)) + (pi)(2)(2sqrt(2)) #
# \ \ \ \ \ =5 sqrt(2) \ pi#