CBSE Guess > Papers > Important Questions > Class XII > 2010 > Maths > Mathematics By Mr. M.P.Keshari

Application of Integrals

8.4. Area of triangle.

Q.1. Using integration, find the area of the triangle ABC, the coordinates of whose vertices are A(2, 0), B(4, 5) and C(6, 3).

Solution :

Fig.

The given points are plotted on a graph as above.
Equation of AB is
y – 0 = {(5 – 0)/(4 – 2)}(x – 2)
=> y = 5/2(x – 2) --------------- (1)
Equation of BC is
y – 5 = {(5 – 3)/(4 – 6)}(x – 4)
=> y = – x + 9 ------------------- (2)
Equation of AC is
y – 0 = {(3 – 0)/(6 – 2)}(x – 2)
=> y = 3/2(x – 2) -----------------(3)
Area of ∆ABC = Area of ∆APB + Area of Trapezium PQCB – Area of ∆AQC
= 2∫4y1dx + 4∫6y2dx – 2∫6y3dx
= 5/2 2∫4(x – 2)dx + 4∫6(9 – x)dx – 3/4 2∫6(x – 2)dx
= 5/2[x2/2 – 2x]24 + [9x – x2/2]46 – 3/4[x2/2 – 2x]26
= 5/2[8 – 8 – (2 – 4)] + [54 – 18 – (36 – 8) – 3/4[18 – 12 – (2 – 4)]
= 5 + 8 – 6
= 7 sq. units. [Ans.]

Q.2. Using integration find the area of the triangular region whose vertices are (1, 0), (2, 2) and (3, 1).

Solution :

Do yourself. [Ans. = 3/2]

Paper By Mr. M.P.Keshari
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