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

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  • Equation of a line when two points are given is $\large\frac{y-y_1}{y_2-y_1}=\frac{x-x_1}{x_2-x_1}.$
Step 1:
Let the points be $A(1,3),B(2,5),C(3,4)$
The equation of the line AB is $\large\frac{y-y_1}{y_2-y_1}=\frac{x-x_1}{x_2-x_1}.$
(i.e) $\large\frac{y-3}{5-3}=\frac{x-1}{2-1}.$
$\Rightarrow y-3=2x-2$
$\Rightarrow y=2x+1$-----(1)
The equation of the line BC is
(i.e) $\large\frac{y-5}{4-5}=\frac{x-2}{3-2}.$
$\Rightarrow y-5=-1(x-2)$
$\Rightarrow y=7-x$------(2)
The equation of the line AC is
(i.e) $\large\frac{y-3}{4-3}=\frac{x-1}{3-1}.$
$\Rightarrow2( y-3)=x-1$
Step 2:
Area of the required region is
On integrating we get,
Step 3:
On applying limits we get,
On simplifying we get,
answered Oct 2, 2013 by sreemathi.v

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