Find the volume?

Question

Find the volume of the solid ball x^2+y^2+z^2<=1.

Answer 1

Who not use spherical coordinates?

volume of the solid ball

= ∫dr ∫r sinΦ dθ ∫r dφ, where r is from 0 to 1, θ is from 0 to 2pi, and φ is from 0 to pi

= 4pi∫{0, 1} r^2 dr

= (4/3)pi <==Answer

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Attn: dr, r sinΦ dθ, and r dφ are the three dimensions of the cell volume.

Answer 2

You Are probably being asked by your professor to find it with calculus.

Here is the calculation for any radius R.

divide it into horizontal disk cross sections.

At height z, the cross section has radius √(R²-z²)

The area of the section is therefore π*(R² - z²).

Integrating for z in [-R,R] gives the volume.

V = π∫_-R^R (R² - z²) dz

= π[R²z - z³/3]_-R^R

= π[(2R³/3)-(-2R³/3)]

= 4πR³/3

With R = 1 you get 4π/3.

Answer 3

Hi..

The equation x^2+y^2+z^2<=1. represents a sphere of radius (actually less that on equal to 1) The general equation of a sphere whose center is at origin is x^2+y^2+z^2 = r^2 where is the radius. So in your equation you know tat the sphere has radius r<=1 so the volume of a sphere is V= 4/3 pi r^3. If you plug in r=1 we get V = 4/3 Pi where Pi = 3.14 (approx_ so the volume of your ball here is <= 4/3 Pi cubic units,,hope this helps!