Showing posts with label how to find area under curve. Show all posts
Showing posts with label how to find area under curve. Show all posts

Find two positive numbers whose sum is 16 and sum of whose cube is Minimum


Show that of all the rectangles inscribed in a circle of given radius . The Square has maximum Area.


Solutions


Let ABCD be rectangle which is  inscribed in a given circle of radius ‘r’
Show that of all the rectangles inscribed in a circle of given radius . The Square has maximum Area.
And Let θ be the angle between side of rectangle and Diameter of given circle.


Therefore from right angled  Δ ABC ,

We have 
  AB  = AC cosθ          ∵ AC = 2r
Let A(x) be the area of Rectangle ABCD
∴ A(x) = AB × BC
    A = (2r cos θ)(2r sin θ )
    A =  4r2 sin θ cos θ
    A = 2r2  (2sin θ cos θ)
    A = 2r2  (sin 2θ )

⇒ 2r2 2 (cos 2θ ) = 0 ,As r2 is constant
⇒cos 2θ = 0
⇒cos 2θ =cos (π/2)
⇒ θ = π/4
 =4r2  (-2sin 2θ 



∴ A has Maximum value at θ = π/4



Find two positive numbers whose sum is 16 and sum of whose cube is Minimum

Solution

Let us consider two numbers x and 16- x .
Then transforming our problem to mathematical form which says “sum of whose cube”  as follows
A (x) =   x3 + (16 - x)3…….. (1)
Differentiating both sides w .r. t  “x” , we get



     X = 8
So  x  =  8 will be the 1st required numbers if Double derivatives of A  w. r. t  ‘x’ comes to be positive at x = 8.
Differentiate (2)  w. r. t. ‘x’  .




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HOW TO FIND AREA BOUNDED BY TWO CIRCLES , INTEGRATION OF AREA UNDER CURVE

HOW TO FIND AREA OF TWO CIRCLES INTERSECTING EACH OTHERS,

Here we are going to discuss how to find common area of two circles which are overlapping or intersecting at two points with the help of an example
 Let us consider two circles whose equations are  given below
x2 + y2 =  12                   .......................(1)
x - 1 )2 y2 12          .....................(2)


HOW TO FIND AREA  BOUNDED BY  TWO CIRCLES Let us draw these circles in coordinate planes, We can compare these equations with standard form of circle to find  the coordinate of  centre of both the circles are (0,0) and (1,0) respectively and radius of both the circles are 1.


If these two circles intersect with each other then we have to find their point/s of intersection.
To find points of intersection subtracting equation (1) from eq (2) , we get 

    X - 1 )2 Y2 - X2 Y=  12 - 1
⇒( X - 1 )2  - X2 =  0
 X 2  +1 2  - 2×(1)×(X) X2 =  0


⇒1 - 2X = 0
⇒ X = 1/2 ,
Now to find the values of y put the value of x in equation # 1
(1/2)2 Y=  12     
   Y1- (1/4) = 3/4

   Y = 土⇃(3/4)
Therefore two points of intersection are B(1/2 , ⇃(3/4)) , C ( (1/2 , -⇃(3/4))

To understand better the solution of  this problem watch this  video 

Required area = shaded  Area , 
we can divide shaded area into four equal parts , As each parts is symmetrical , Therefore to find shaded area  it is sufficient to find the area of any one of four part and then then multiply it with 4.

Hence  Required area = 4 area OBLO = 4 Area BALB
To avoid tedious calculations choose 2nd part to integrate
I= Required area = 4 Area BALB    ------------- (3)
HOW TO FIND AREA  BOUNDED BY  TWO CIRCLES
After simplification , we have



    ALSO READ        HOW TO INTEGRATE INTEGRAL WITH SQUARE ROOT IN NUMERATOR     

My previous post HOW TO FIND AREA OF THE CIRCLE  WHICH IS INTERIOR TO THE PARABOLA

  Final words 


Thanks for visiting this website and spending your valuable time to read this post regarding how to find area bounded by two circles .If you liked this post , do share it with your friends to benefit them also we shall meet in next post , till then bye and take care....

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HOW TO FIND AREA OF THE CIRCLE WHICH IS INTERIOR TO THE PARABOLA


HOW TO FIND AREA OF THE CIRCLE  WHICH IS INTERIOR TO THE PARABOLA

Area Under Curves

Let  us write two equations of circle and parabola respectively
4x2+ 4y2  = 9 ------------------- (1) 

and x2      = 4y    -------------------(2)

Reducing (1) to standard form by dividing 4 .we get 
x2y2  (3/2)2   
Ist of all draw figures of both the circle and the parabola in cartesian plane.


As it can be seen from figure both  curves intersect each other at two points say A and A' . 
Next we have to find these two coordinates points of intersection . Solving (1) and (2) to find the values of x and y 
Putting  the value of ' x2 from (2) in (1) we get 

4(4y)+ 4y2 = 9

16y + 4y2 - 9 = 0

 4y2 - 16y -9 = 0
        
 y = (-16+20)/8  and (-16-20)/8
y = 1/2  and -9/2


HOW TO FIND AREA OF THE CIRCLE  WHICH IS INTERIOR TO THE PARABOLA

So Rejecting the -ve value of y ,because when we put negative value (-9/2) in eq (2) , we shall have two complex values of "x" which are not acceptable.
so only put positive value (1/2) of   'y'  in (2) we get two real values of  'x' such that    x= 土⇃2,
Now we can write coordinate M(⇃2,0) and N (-⇃2,0)

Required Area = Shaded area
                         = 2 × Area OBAO 
Note this step carefully


                   
HOW TO FIND AREA OF THE CIRCLE  WHICH IS INTERIOR TO THE PARABOLA
Multiplying every terms with 2 which is written at  beginning  of the previous line.


HOW TO FIND AREA OF THE CIRCLE  WHICH IS INTERIOR TO THE PARABOLA

Putting the values of upper and lower limits of x
HOW TO FIND AREA OF THE CIRCLE  WHICH IS INTERIOR TO THE PARABOLA


    ALSO READ        HOW TO INTEGRATE INTEGRAL WITH SQUARE ROOT IN NUMERATOR     


  Final words 


Thanks for visiting this website and spending your valuable time to read this post.If you liked this post , do share it with your friends to benefit them also we shall meet in next post , till then bye and take care......

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