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.
Integration exercises solution 2 :
We notice that the function f is positive on [0;ㄫ/2].
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 :
then: V= 1/4 π² u.v
Find the original function :
In these exercises we will use the integral calculus to find the original function of the function we want to integrate.
Determine the sphere volume formula :
Using integral calculus to find the volume of a 3D stereoscopic is like the first exercise which we found the formula for the volume of a sphere.
Also, we will calculate the volume of a stereoscope generated by a curve rotation around an axis.
Determine the area of a plane :
Finding the area of the plane intertwined between the curve and the x axis, as in the second exercise.