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प्रश्न
Find the area of the region included between y = x2 + 5 and the line y = x + 7
उत्तर
Given equation of the curve is
y = x2 + 5 ......(i)
and equation of the line is
y = x 7 ......(ii)
Find the points of intersection of y = x2 + 5 and y = x + 7
Substituting (ii) in (i), we get
x + 7 = x2 + 5
∴ x2 – x – 2 = 0
∴ (x – 2)(x + 1) = 0
∴ x = 2 or x = –1
When x = 2, y = 9 and when x = –1, y = 6
∴ The points of intersection are B(2, 9) and A(–1 , 6).
Draw BD ⊥ OX.
Required area = area of the region AEBCA
= area of the region FODBCAF − area of the region FODBEAF
= area under the line y = x + 7 – area under the curve y = x2 + 5
= `int_(-1)^2 (x + 7) "d"x - int_(-1)^2 (x^2 + 5) "d"x` .....[From (ii) and (i)]
= `int_(-1)^2 (x + 7 - x^2 - 5) "d"x`
= `int_(-1)^2 (x - x^2 + 2) "d"x`
= `[x^2/2]_(-1)^2 - [x^3/3]_(-1)^2 + 2[x]_(-1)^2`
= `1/2[2^2 - (-1)^2] - 1/3[2^3 - (-1)^3] + 2[2 - (-1)]`
= `3/2 - 9/3 + 6`
= `9/2` sq.units
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