Find the area of the region bounded by the curve $x = a t^{2}, y = 2 a t$ between the ordinates corresponding t = 1 and t = 2. [NCERT EXEMPLAR]

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Solution

The curve $x = a t^{2}, y = 2 a t$ represents the parametric equation of the parabola.

Eliminating the parameter t, we get

$y^{2} = 4 a x$

This represents the Cartesian equation of the parabola opening towards the positive x-axis with focus at (a, 0).

When t = 1, x = a

When t = 2, x = 4a

∴ Required area = Area of the shaded region

= 2 × Area of the region ABCFA

$= 2 \int_{a}^{4 a} y_{parabola} d x = 2 \int_{a}^{4 a} \sqrt{4 a x} d x = 2 \times {2 \sqrt{a} \times \frac{x^{\frac{3}{2}}}{\frac{3}{2}}}_{a}^{4 a} = \frac{8 \sqrt{a}}{3} [{(4 a)}^{\frac{3}{2}} - a^{\frac{3}{2}}] = \frac{8 \sqrt{a}}{3} (8 a \sqrt{a} - a \sqrt{a}) = \frac{8 \sqrt{a}}{3} \times 7 a \sqrt{a} = \frac{56}{3} a^{2} square units$

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Similar questions

x = a t^{2}

y = 2 a t

t = 1

t = 2

y = \sqrt{x}

y = \sqrt{x}, x = 2 y + 3

x = a t^{2}, y = 2 a t

1 \leq t \leq 3

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