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Question
Find the area of the region bounded by the curve y = x3 and y = x + 6 and x = 0
Solution
We are given that: y = x3, y = x + 6 and x = 0
Solving y = x3 and y = x + 6
We get x + 6 = x3
⇒ x3 – x – 6 = 0
⇒ x2(x – 2) + 2x(x – 2) + 3(x – 2) = 0
⇒ (x – 2)(x2 + 2x + 3) = 0
x2 + 2x + 3 = 0 has no real roots.
∴ x = 2
∴ Required area of the shaded region
= `int_0^2 (x + 6) "d"x - int_0^2 x^3 "d"x`
= `[x^2/2 + 6x]_0^2 - 1/4 [x^4]_0^2`
= `(4/2 + 12) - (0 + 0) - 1/4 [(2)^4 - 0]`
= `14 - 1/4 xx 16`
= 14 – 4
= 10 sq.units
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