Let S be the sphere of radius R. a. Argue by symmetry that

∫ ∫ x^2dS = ∫ ∫ y^2 dS= ∫ ∫ z^2dS
S S S

b. ) Use this fact and some clever thinking to evaluate, with very little computation, the integral

∫∫x^2 dS.
S

Step-by-step explanation:

a. Given that the sphere is symmetrical in nature. If we take a look at the yz-plane, over the hemisphere,  the integral on the surface of x on both sides of the plane will annul each other and the outcome will be zero. However, the remaining two integrals will also be zero as a result of symmetry on the xy and xz planes.

Thus,

b. Recall that :

By applying the integral of the surface area.

      (surface area of a sphere)

From above;

∴

a

step-by-step explanation:

answer: (yeees, you need a poet.)

//blooming serenity//

beautiful in the morning

mysterious at night

wonderous nature souring

with all it's mystical might

the flower that blooms

to the fox that looms

nocturnal creatures

with various features

a blooming flower soon in sight

as a new little hatchling fears to take flight  

afraid to leave it's calming nest,

but dear little bird, mother knows best

as sure as there are stars in the sky,

life is not about who, where what, when or why-

it's the journey that follows

adventures we swallow

of   nature's deep wonders,

natural disasters and storm's thunders.

blooming serenity

life and nature, truly is heavenly.

-kerstien lehman

step-by-step explanation: