The region under the curve #y=sqrt(x^2-4)# bounded by #2<=x<=4# is rotated about a) the x axis and b) the y axis. How do you sketch the region and find the volumes of the two solids of revolution?

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Answer 1 · 2017-07-27T20:52:54

Please see below.

Here is the region:

enter image source here

About #x# axis

Use discs.

#V = int_2^4 pir^2 dx = int_2^4 pi(sqrt(x^2-4))^2 dx#

# = pi int_2^4 (x^2-4) dx#

# = pi[x^3/3-4x}_2^4#

# = (32pi)/3#

About #y# axis

Use shells.

#V = int_2^4 2pirh dx = 2pi int_2^4 xsqrt(x^2-4) dx#

Integrate by substitution #u = x^2-4#

# = 2pi [1/3(x^2-4)^(3/2)]_2^4#

# = 16sqrt3 pi#