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Question

Find the area of region bounded by the triangle whose vertices are (1,0),(2,2),(3,1) using integration.

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Solution

Area of ABC= Area of ABD+ Area of trapezium BDEC Area of ACE
Area ABD
Area ABD =21ydx
Equation of line between A(1,0) B(2,2) is
y0x1=2021

yx1=2

y=2(x1)
Area ABD =212(x1)dx
=2[x22x]21
=2[2212+1]
=1

Area BDEC
Area BDEC =32ydx
Equation of line between B(2,2) & C(3,1) is
y2x2=1232
y2=(x2)
y=4x
Area BDEC =32(4x)dx
=[4xx22]32
=4(32)12(3222)
=412×5
=32

Area ACE
Area ACE =31ydx
Equation of line between A(1,0) & C(3,1) is
y0x1=1031
yx1=12
y=12(x1)
Area ACE =3112(x1)dx
=12[x22x]31
=12[92312+1]
=1
Hence,
Area of ABC= Area of ABD+ Area of trapezium BDEC Area of ACE
=1+321
=32

1101445_1198704_ans_63704dccebbe4397804cb04ad8e388d7.png

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