Finding volume triple integralFind the volume of the solid enclosed on the outside of the sphere r = 2 and on the inside by the surface r = 1 + cos ⁡ ϕ.

Question

Finding volume triple integralFind the volume of the solid enclosed on the outside of the sphere   r  =  2 and on the inside by the surface   r  =  1  +  cos  ⁡  ϕ.

Jamir Young · Accepted Answer

Step 1Since       d    3        x    =      r    2    d  r  sin  ⁡  θ  d  θ  d  ϕ, your integral is  V  :=      ∫    0    π    sin  ⁡  θ  d  θ      ∫    0          2      π        d  ϕ      ∫          1      +      cos      ⁡      ϕ        2        r    2    d  r  .Step 2Since odd powers of   cos  ⁡  ϕ will integrate to 0,  V  =      2    3        ∫    0          2      π        (  7  −  3      cos    2    ⁡  ϕ  )  d  ϕ  =            22      π        3    .

Find the volume of the solid enclosed on the outside of the sphere r=2 and on the inside by the surface r=1+cos phi.

Open question

Answer & Explanation

New Questions in High school geometry