Find the volume of the solid cut from the thick-walled cylinder 1 less than or equals x squared plus y squared less than or equals 2 by the cones z equals plus or minus StartRoot 4 x squared plus 4 y squared EndRoot.

Question

contestada

Respuesta :

kalsoomtahir1 kalsoomtahir1 · Answer 1 · 2019-12-08T18:30:20+01:00

Answer:

[tex]4/3(2\sqrt{2} -1)*\pi[/tex] this is the answer

Step-by-step explanation:

[tex]1<= x^2+y^2 <= 2 by the cons z= +- \sqrt{(x^2 + y^2)}[/tex]

Let suppose

x = r∙cos(θ)

y = r∙cos(θ)

z = z

The differential volume element changes to

dV = dxdydz = r drdθdz

The limits of integration in cylindrical coordinates are:

(i)

[tex]1 \leq x^2 + y^2\leq 2[/tex]

[tex]1 \leq r^2 \leq 2[/tex]

since r is always positive

[tex]1 \leq r \leq \sqrt{2}[/tex]

(ii)

[tex]- \sqrt{(x^2+ y^2)} \leq z \leq +\sqrt{(x^2+ y^2)}[/tex]

[tex]- r \leq z \leq r[/tex]

(iii)

[tex]0 \leq Theta \leq 2∙π[/tex]

we have no restrictions in radial direction.

[tex]V = \int\limits^a_b { dV} \,[/tex]

Remaining derivation has been explained in the atatchment where we get the volume of the cylinder