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Integration exercises 1 : ( sphere volume  formula )

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

                                                       =43πR3


Integration exercises solution 1 : 

Cartesian coordinate system for a three-dimensional space




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:


Integration exercises with answers

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.

f : x ⟶ cos x  on the domain  [0;ㄫ/2]

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


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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