Q:

# 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.

Accepted Solution

A:
Answer:$$4/3(2\sqrt{2} -1)*\pi$$ this is the answerStep-by-step explanation: $$1<= x^2+y^2 <= 2 by the cons z= +- \sqrt{(x^2 + y^2)}$$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: The limits of integration  in cylindrical coordinates are:   (i) $$1 \leq x^2 + y^2\leq 2$$ $$1 \leq r^2 \leq 2$$ since r is always positive $$1 \leq r \leq \sqrt{2}$$ (ii) $$- \sqrt{(x^2+ y^2)} \leq z \leq +\sqrt{(x^2+ y^2)}$$ $$- r \leq z \leq r$$(iii)   $$0 \leq Theta \leq 2∙π$$  we have no restrictions in radial direction.$$V = \int\limits^a_b { dV} \,$$ Remaining derivation has been explained in the atatchment where we get the volume of the cylinder