Finding volume using triple integral when i have given set.Find the volume of: V = [ ( x , y , z ) : 0 ⩽ z ⩽ 4 − x 2 + y 2 , 2 x ⩽ x 2 + y 2 ⩽ 4 x ]I should somehow construct triple integral here in order to solve this, which means that i have to find limits of integration for three variables, but i am just not quite sure how, i assume that i should first integrate for z since we could say that limits for z are already given, but what i am supposed to do with other two variables. When i find limits, what function i am going to integrate, is it going to be just ∭ d z d y d x?

Question

Finding volume using triple integral when i have given set.Find the volume of:  V  =  [  (  x  ,  y  ,  z  )  :  0  ⩽  z  ⩽  4  −            x      2        +          y      2        ,  2  x  ⩽      x    2    +      y    2    ⩽  4  x  ]I should somehow construct triple integral here in order to solve this, which means that i have to find limits of integration for three variables, but i am just not quite sure how, i assume that i should first integrate for z since we could say that limits for z are already given, but what i am supposed to do with other two variables. When i find limits, what function i am going to integrate, is it going to be just   ∭  d  z  d  y  d  x?

Chaya Garza · Accepted Answer

Explanation:  2  x  ≤      x    2    +      y    2   is the same as   0  ≤      x    2    −  2  x  +      y    2    =      x    2    −  2  x  +  1  −  1  +      y    2    =  (  x  −  1      )    2    +      y    2    −  1 or   (  x  −  1      )    2    +      y    2    ≥  1. That is the set of points outside the circle with center at (1, 0) and radius 1. Similarly,       x    2    +      y    2    ≤  4  x is the same as       x    2    −  4  x  +      y    2    =      x    2    −  4  x  +  4  −  4  +      y    2    ≤  0 or   (  x  −  2      )    2    +      y    2    ≤  4, the interior of the circle with center at (2, 0) and radius 2.

Leia Hood · Accepted Answer

Step 1Whenever you see that a volume is defined by having a             a      2        +          b      2       for some variables a and b you must immediately consider using cylindrical/spherical coordinates. In your case you have a             x      2        +          y      2       so you should immediately consider  x  =  ρ  cos  ⁡      θ    y  =  ρ  sin  ⁡      θ    z  =  zStep 2Substituting in the expression for your volume we get  V  =  [  (  ρ  ,  θ  ,  z  )  :  0  ⩽  z  ⩽  4  −  ρ  ,  2  cos  ⁡      θ    ⩽  ρ  ⩽  4  cos  ⁡      θ    ]Because   θ has no imposed restrictions you get   0  ⩽  θ  ⩽  2  π; After that you get   2  cos  ⁡      θ    ⩽  ρ  ⩽  4  cos  ⁡      θ   and then   0  ⩽  z  ⩽  4  −  ρ. With this you should be able to build your triple integral with the new coordinates and compute it.

Find the volume of: V=[(x,y,z):0leqzleq4-sqrt(x^2+y^2),2xleqx^2+y^2leq4x]

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