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प्रश्न
Find the area of the region bounded by the curve 2 + x – x2 + y = 0, x axis, x = – 3 and x = 3
उत्तर
Given curve is
2 + x – x2 + y = 0
y = x2 – x – 2
y = (x – 2)(x + 1)
Area required = `int_(-3)^(-1) (x^2 - x - 2) "d"x + |int_(-1)^2 (x^2 - x - 2) "d"x| + int_2^3 (x^2 - x - 2) "d"x`
= `[x^3/3 - x^2/2 - 2x]_(-3)^(-1) + |[x^3/3 - x^2/2- 2x]_(-1)^2| + [x^3/3 - x^2/2 - 2x]_2^3`
= `[- 1/3 - 1/2 + 2 + 9 + 9/2 - 6] + |[8/3 - 4/2 - 4 + 1/3 + 1/2 - 2]| + [27/3 - 9/2 - 6 - 8/3 + 4/2 + 4]`
= `26/3 + 9/2 + 11/6`
= `(52 + 24 + 11)/6`
= `90/6`
= 15 sq.units
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