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Question
Find the area between the two curves (parabolas)
y2 = 7x and x2 = 7y.
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Solution
To find the point of intersection of the curves:
Equation of curve is x2 = 7y
∴ y = `x^2/7`
y2 = `x^4/49` ......(1)
Equation to second curve,
y2 = 7x ......(2)
Equating equations (1) and (2) we get
`x^4/49` = 7x
⇒ x4 = 343x
⇒ x4 – 343x = 0
⇒ x(x3 – 343) = 0
⇒ x = 0
or x3 = 343
⇒ x = 7
When x = 0, y = 0
When x = 7, y = 7
∴ The points of intersection of parabolas are (0, 0) and (7, 7).
∴ Required area, A = `|int_0^7 y_1 . dx - int_0^7 y_2. dx|`
= `|int_0^7 sqrt(7x) . dx - int_0^7 x^2/7. dx|`
= `|sqrt(7) [x^(3/2)/(3/2)]_0^7 - 1/7 [x^3/3]_0^7|`
= `|2/3 xx sqrt(7) xx 7^(3/2) - 1/21 7^3|`
= `|2/3 xx 7^2 - 1/21 xx 7^3|`
= `|2/3 xx 7^2 - 1/3 xx 7^2|`
= `7^2(2/3 - 1/3)`
= `49/3` sq.units
Hence area between the two curves is `49/3` sq.units.
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