Question

Using integration, find the area of the triangular region whose sides lie along the lines y=2x+1,y=3x+1 and x=4.

Answer 1

A bag contains Rs.396 in terms of 50paise, 2rupee and 5rupee coins in the ratio 4:3:2. Find the number of each type of coins

Answer 2

The area of a triangle thus formed is 8 square units.

Step-by-step explanation:

A triangle is enclosed by the 3 lines

y+2x+1−−−−(1)

y=3x+1−−−−(2)

x=4 −−−−−−(3)

To get the vertices of the triangle, we need to solve the 3 equations.

To find the vertex A, let us solve equations (1) & (2):

y=2x+1 and y=3x+1

So 2x+1 = 3x+1. We get x = 0 and y = 1

Hence vertex A is (0,1)

To find the vertex B let us solve equations (2) & (3):  

y=3x+1 and x=4

Substituting x, we get y = 12+1 = 13

So vertex B is (4,13)

To find the vertex C, let us solve equations (3) & (1):

x=4 and y=2x+1

Substituting x, we get y = 8+1 = 9

So vertex C is (4,9)

Please refer to the attached picture for integration and diagram.

On applying limits we get,

Area = (24+4)−(16+4)

       =28−20

       =8 square units

The area of a triangle thus formed is 8 square units.