math exercises integration
Integration exercises 1 : ( sphere volume formula )
Prove that the volume of a sphere its radius R is:
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:
4/3 ㄫ R3
Integration exercises 2:
Let (C) The graph of the function f : x ⟶ cos x on the domain
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.