math exercises integration
Integration exercises 1 : ( sphere volume formula )
Prove that the volume of a sphere its radius R is:
∨=43Ï€R3
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 ã„« R3
Integration exercises 2:
Let (C) The graph of the function f : x ⟶ cos x on the domain
[0;ã„«/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;ã„«/2].
1)
A=∫Ï€20cos(x)dx
A=[sin(x)]Ï€20
A=1 u.a
2)
V=∫Ï€20Ï€[f(x)]2dx
V=Ï€∫Ï€20cos2(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=Ï€[12(x+12sin(2x))]Ï€20=12(Ï€2−0)=Ï€24
then: V= 1/4 Ï€² 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.
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