Question

Find the volume bounded by the paraboloid 𝒙^𝟐+𝒚^𝟐=𝒂𝒛 and the cylinder 𝒙^𝟐+𝒚^𝟐=𝒂^𝟐. 

 

Answer 1

The equations of the cylinder and the paraboloid in polar form are r = a and r² = az. 

Now, z varies from z = 0 to z = r²/a, r varie from r = 0 to r = a and θ varies from θ = 0 to θ =` pi/2` taken 4 times. 

 

∴ `V=4int_(θ=0)^(pi/2) int_(r=0)^a int_(z=0)r^2/a`  𝑑𝑟 𝑑𝜃 𝑑𝑧 

∴` V=4 int_(θ=0)^(pi/2) r[Z]_0^((r"^2)/a)`  𝑑𝑟𝑑𝜃 

∴ `V= 4 int_(θ=0)^(pi/2) int_(r=0)^a r^3/a dr dθ` 

∴ `V=4/a int_θ^(pi/2) [r^4/4]_0^a dθ` 

∴` V=4/a int_0^(pi/2) a^4/4 dθ` 

∴ `V=a^3 int_0^(pi/2) dθ`  

∴` V= a^3 [θ]_0^(pi/2)` 

∴ `V=( pia^3)/2`