Question

find the volume lying above the xy plane inside the cylinder x^2+y^2=4 and under the cone x^2+y^2=z^2​

Answer 1

QUESTION

find the volume lying above the xy plane inside the cylinder x^2+y^2=4 and under the cone x^2+y^2=z^2​

$\boxed{\huge{\sf{Answer}}}$

Suppose that g(x,y,z) is continuous on a rectangular box B, which when described in cylindrical coordinates looks like B={(r,θ,z)|a≤r≤b,α≤θ≤β,c≤z≤d}.

Then g(x,y,z)=g(rcosθ,rsinθ,z)=f(r,θ,z) and

∭B g(x,y,z)dV=  d

∫

c  

β

∫

α  

b

∫

a f(r,θ,z)rdrdθdz.

$\mathfrak{\underline{\underline{\red{Given: }}}}$

If the function f(r,θ,z) is continuous on B and if (r

*

ijk

,θ

*

ijk

,z

*

ijk

) is any sample point in the cylindrical subbox Bijk[ri−1,ri]×[θj−1,θj]×[zk−1,zk] (Figure 5.51), then we can define the triple integral in cylindrical coordinates as the limit of a triple Riemann sum, provided the following limit exists:

liml,m,n→∞  

l

∑

i=1  

m

∑

j=1  

n

∑

k=1 f(r

*

ijk

,θ

*

ijk

,z

*

ijk

)r

ijk

ΔrΔθΔz.

$/\boxed{Hope /; its\;helps /;you}$