CBSE Class 12 · 2012 · Area Under CurvesLong Answer

Using the method of integration, find the area of the region bounded by the following lines:

5 x - 2 y - 10 = 0

x + y - 9 = 0

and

2 x - 5 y - 4 = 0

Answer: $\frac{21}{2}$ square units

Solution

Solving the lines in pairs gives the vertices

(2, 0)

(lines 1 and 3),

(4, 5)

(lines 1 and 2) and

(7, 2)

(lines 2 and 3).
Writing

y

for each line (line 1:

y = \frac{5 x - 10}{2}

, line 2:

y = 9 - x

, line 3:

y = \frac{2 x - 4}{5}

) with the lowest line

2 x - 5 y - 4 = 0

as the lower edge:

A = \int_{2}^{4} \frac{5 x - 10}{2} d x + \int_{4}^{7} (9 - x) d x - \int_{2}^{7} \frac{2 x - 4}{5} d x .

= 15 - 10 - \frac{33}{2} + 27 - 9 + 4 = 27 - \frac{33}{2} = \frac{21}{2} .

Hence the area of the triangular region is

\frac{21}{2}

square units.

CBSE Class 12 Board · 2012 · Subjective