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?

Question

3567 views around the world

You can reuse this answer
Creative Commons License

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#