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

Bobby Mitchell

Bobby Mitchell

Answered question

2022-08-07

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?

Answer & Explanation

Chaya Garza

Chaya Garza

Beginner2022-08-08Added 10 answers

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

Leia Hood

Beginner2022-08-09Added 2 answers

Step 1
Whenever 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 = z
Step 2
Substituting 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.

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