Question

Find the area between the two curves (parabolas)

y² = 7x and x² = 7y.

Answer 1

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.