Q26 of 45 Page 1

Using integration, find the area of the region bounded by the triangle whose vertices are (-1, 2), (1, 5) and (3, 4).

Plot these points and form a triangle



Write the equations of line AB, BC and AC using two-point form


Equation of line AB,


A = (-1, 2) = (x1, y1) and B = (1, 5) = (x2, y2)




⇒ -3(x + 1) = -2(y – 2)


⇒ 3x + 3 = 2y – 4


⇒ 2y = 3x + 7


⇒ y = 1/2(3x + 7)


Equation of line BC,


B = (1, 5) = (x1, y1) and C = (3, 4) = (x2, y2)




⇒ x – 1 = -2(y – 5)


⇒ x – 1 = -2y + 10


⇒ -2y = x – 11


⇒ y = -1/2(x – 11)


Equation of line AC,


A = (-1, 2) = (x1, y1) and C = (3, 4) = (x2, y2)




⇒ -2(x + 1) = -4(y – 2)


⇒ x + 1 = 2(y – 2)


⇒ x + 1 = 2y – 4


⇒ 2y = x + 5


⇒ y = 1/2(x + 5)


Now the area of triangle = area under AB + area under BC – area under AC as shown …(i)



Let us find the area under AB


⇒ y = 1/2(3x + 7)


Integrate from -1 to 1










⇒ area under AB = 7 unit2 …(a)


the area under BC,


⇒ y = -1/2(x – 11)


Integrate from 1 to 3











⇒ area under BC = 9 unit2 …(b)


the area under AC,


⇒ y = 1/2(x + 5)


Integrate from -1 to 3











⇒ area under AC = 7 unit2 …(c)


Using (a), (b) and (c) in (i)


⇒ area of triangle = area under AB + area under BC – area under AC


⇒ area of triangle = 7 + 9 – 7


⇒ area of triangle = 9 unit2


Hence area of triangle is 9 unit2


More from this chapter

All 45 →
24

Show that the altitude of the right circular cone of maximum volume that can be inscribed in a sphere of radius r is Also show that the maximum volume of the cone is of the volume of the sphere.

 25

Evaluate:

27

Find the equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x - y + z = 0. Also find the distance of the plane obtained above, from the origin.

OR


Find the distance of the point (2, 12, 5) from the point of intersection of the line and the plane

28

A manufacturing company makes two types of teaching aids A and B of Mathematics for class XII. Each type of A requires 9 labour hours of fabricating and 1 labour hour for finishing. Each type of B requires 12 labour hours for fabricating and 3 labour hours for finishing. For fabricating and finishing, the maximum labour hours available per week are 180 and 30 respectively. The company makes a profit of Rs80 on each piece of type A, and Rs120 on each piece of type B. How many pieces of type A and type B should be manufactured per week to get a maximum profit? Make it as an LPP and solve graphically. What is the maximum profit per week?