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Question
Sketch the graph of y = |x + 4|. Using integration, find the area of the region bounded by the curve y = |x + 4| and x = –6 and x = 0.
Solution
y = x + 4, if x > 4 and y = -(x+4), if x < 4
For y = x + 4
when x = 0, y= 4
and when y = 0, x = -4
Point are
∴ (0,4) and (-4,0)
For y = -x - 4
when x = 0, y = -4
when y = 0, x= -4
∴ Point are
(0, -4) and (-4,0)
∴ Required area
`= int_(-6)^(-4)-(x+4) dx + int_(-4)^0 (x + 4) dx`
`= -[x^2/2 + 4x]_(-6)^(-4) + [x^2/2 + 4x]_(-4)^0`
`= [(-4)^2/4 + 4(-4) - [(-6)^2/2 + 4(-6)]] + [0 + 0[(-4)^2/2 + 4(-4)]]`
`= -[16/2 - 16 -[36/2 - 24]] + [-(16/2 - 16)]`
= -[-8 + 6] + [8]
2 + 8 = 10 sq. unit
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