## math exercises integration

## Integration exercises 1 : ( sphere volume formula )

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

**∨****=43π****R****3**

## 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 R**^{3}## 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=π∫π20cos2(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(π2−0)=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.

#### 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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