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प्रश्न
Find the area of the region enclosed by the parabola x2 = y and the line y = x + 2
उत्तर
Here, x2 = y and y = x + 2
∴ x2 = x + 2
⇒ x2 – x – 2 = 0
⇒ x2 – 2x + x – 2 = 0
⇒ x(x –2) + 1(x – 2) = 0
⇒ (x – 2)(x + 1) = 0
∴ x = –1, 2
Graph of y = x + 2
| x | 0 | –2 |
| y | 2 | 0 |
Area of the required region
= `int_(-1)^2 (x + 2)"d"x - int_(-1)^2 x^2 "d"x`
= `[x^2/2 + 2x]_(-1)^2 - 1/3[x^3]_-1^2`
= `[(4/2 + 4) - (1/2 - 2)] - 1/3 [8 - (-1)]`
= `(6 + 3/2) - 1/3(9)`
= `15/2 - 3`
= `9/2` sq.units
Hence, the required area = `9/2` sq.units
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