Integration exercises 1 : ( sphere volume  formula )

Prove that the volume of a sphere its radius R is:

=43πR3

Integration exercises solution 1 :

Consider a Cartesian coordinate system for a three-dimensional space  Its axes
The sphere is its center O And radius R.
Clip this ball with a plane parallel to the plane (xoy) and Z-axis where  -R<Z<R  is a sphere  ts center Ω (0;0;Z) and its radius r=ΩM with OM=R.
We have it in the right triangle OΩM: r²=R²-Z²
And from it the disk area in which its center Ω And its  radius R  is:

then:                    V=
/
4/3 ㄫ R3

Integration exercises 2:

Let (C) The graph of the function  f : x ⟶ cos x  on the domain
[0;ㄫ/2].
1- Calculate A  the area of the plane determined by the curve (C) and the x-axis (x'x).
2- Calculate V the generated volume by rotating the curve around the x-axis.

Integration exercises solution 2 :

We notice that the function f is positive on [0;ㄫ/2].
1)
A=π20cos(x)dx
A=[sin(x)]π20

A=1 u.a

2)
V=π20π[f(x)]2dx

V=ππ20cos2(x)dx

We know that  cos²(x)= (1+cos(2x))/2 , And from it the original function of the function  x ⟶ cos²(x) is the function
x ⟶ 1/2 [ x+ 1/2 sin (2x)]
And so we find volume V :

V=π[12(x+12sin(2x))]π20=12(π20)=π24

then:  V= 1/4 π² u.v