USING METHOD OF INTEGRATION ,HOW TO FIND AREA OF TRIANGLE BOUNDED BY THREE LINES

Using method of integration find the area of triangle,using the method of integration find the area of the region bounded by the lines,area of triangle by integration method 

Using Method of Integration , How to find the area of triangle bounded by three lines 

2x + y = 0 , 3x - 2y = 6  and x - 3y  + 5 = 0

Solution

Given lines are

2x + y  = 4   --------  (1)
3x - 2y = 6  ---------  (2) and 
x - 3y  = -5  ---------  (3)


If these lines are intersecting then we have to find their coordinates of points of intersection .

To Find Coordinate of Point A

Multiply (1) by 2 and adding to (2) , we get
4x + 2y + 3x - 2y = 8 + 6
7x = 14 ⇒ x =2 
Putting x = 2 in (1) , we get 

Area under Curve

2(2) + y = 4
4 + y = 4 ⇒ y = 0
∴ (1) and (2) meets at point A(2,0).

To Find Coordinate of Point B

To find point of intersection (2) and (3);
Multiply (3) by -3 and adding to (2) , we get
3x - 2y -3x +9y  = 6 +15 
7y = 21  ⇒ y = 3 

Putting y = 3 in (3) , we get 
x-3(3)  = -5  
⇒ x = -5 +9  ⇒ x = 4 
∴ (2) and (3) meets at point B(4,3).

To Find Coordinate of Point C

To find point of intersection (1) and (3);
Multiply (1) by 3 and adding to (3) , we get
6x + 3y + x - 3y = 12 - 5 
7x = 7  ⇒ x =1 

Putting x = 1 in (1) , we get 
2(1) + y  = 4  
⇒ y =4 - 2  ⇒ y = 2 
∴ (1) and (3) meets at point C(1,2).

we get points of intersection of (1) and (2)  A(2,0),  points of intersection of (2) and (3) B(4,3) and points of intersection of (1) and (3) C(1,2).

Required Area = Shaded Area   =  Area DCBED - Area DCAD -Area ABEA

Also read my previous post  How to find area bounded by two circles 

For better understanding watch this video 

Conclusion 

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