Skip to main content

Integration exercises with answers

 math exercises integration

Integration exercises 1 : ( sphere volume  formula )

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


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 
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].

A=1 u.a



 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.

Read also:


Popular posts from this blog

Tracing shapes for kids ages 3-5 | part 1

Tracing shapes worksheet for preschool Kids will have fun learning their shapes as they trace the shape and color their very own shapes. This is a great activity for toddler, preschool, pre k, kindergarten, and first grade students. Tracing Shapes Worksheets for Preschool  (All Activity Papers) Home Page

Subtraction Word Problems, Worksheet 3

Subtraction Word Problems The child will enjoy solving this subtraction problem by subtracting the broken eggs.

Puzzle of Crossing the Dark Bridge in 17 Minutes

How to cross the bridge in 17 minutes?  4 people who want to cross a bridge at night with a torch have to cross the bridge in the dark in 17 minutes -Since the bridge cannot carry more than two people, it means that the bridge can be crossed by one or two people. -To cross the bridge, a person must be holding the torch in his hand (meaning one person must return the torch so that the others can cross). -The first person A takes 1 minute to cross the bridge. -The second person B takes 2 minutes to cross the bridge. -The third person C takes 5 minutes to cross the bridge. -The fourth person D takes 10 minutes to cross the bridge. How can these people cross the bridge in 17 minutes? The solution: -Person A and Person B cross, so it takes 2 minutes. -Person B returns with the torch so it has taken 2 minutes. -He gives the torch to people C and D to cross t