Compute the volume of E (Calculus 3)....help!?

Question

E = {(x,y,z) E R^(3) | 0
a. Compute the volume of E.

b. triple integral xyz DV

Answer 1

Using Cylindrical Coordinates:



a) The volume is given by

V = ∫(θ = 0 to 2π) ∫(z = 0 to 1) ∫(r = 0 to sqrt(4 - z^2)) 1 * (r dr dz dθ)

...= ∫(θ = 0 to 2π) ∫(z = 0 to 1) r^2/2 {for r = 0 to sqrt(4 - z^2)} dz dθ

...= ∫(θ = 0 to 2π) ∫(z = 0 to 1) (1/2)(4 - z^2) dz dθ

...= 2π * (1/2)(4z - z^3/3) {for z = 0 to 1}

...= 11π/3.



b) ∫∫∫ xyz dV

= ∫(θ = 0 to 2π) ∫(z = 0 to 1) ∫(r = 0 to sqrt(4 - z^2)) (r^2 z cos θ sin θ) * (r dr dz dθ)

= [∫(θ = 0 to 2π) sin θ cos θ dθ] * [∫(z = 0 to 1) ∫(r = 0 to sqrt(4 - z^2)) r^3 z dr dz]

= 0 * [∫(z = 0 to 1) ∫(r = 0 to sqrt(4 - z^2)) r^3 z dr dz]

= 0.



I hope this helps!

Answer 2

this equation you are given looks like the equation of a sphere. I would try using a half circle of radius 2 then spin it around the y-axis and then trying to find the volume using multivariable calculus which I have forgotten. Hope this helps

Answer 3

The fundamental of pi * y^2 dx So this is going to develop into the fundamental from 0 to 3 of x*pi dx = x^2/2 * pi from 0 to 4 = 8pi it is in spite of if this is revolved around the x-axis in spite of if this is revolved around the y-axis the respond is the fundamental of pi * x^2 dy So this is going to develop into the fundamental from 0 to 2 of y^4 * pi dy = y^5/5 * pi = 32/5 * pi