The area bounded by the parabola y² = 4ax and its latus rectum is equal to:

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AAI JE ATC 2016 Shift 1

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Answer 1

Concept:

Area of a region can be calculated by:

\(\int\!\!\!\int dxdy\)

Calculation:

Given:

To find area bounded by parabola y² = 4ax and its latus rectum

Latus rectum of the parabola x = a

RRB ALP Maths FT1 2

\(Area=\int\!\!\!\int dydx\)

\(Area = 2\mathop \smallint \limits_0^a \smallint \limits_0^y \;dydx\)

\(Area = 2\mathop \smallint \limits_0^a y\;dx\)

\(= 2\mathop \smallint \limits_0^a \sqrt {4ax} \;dx\)

\(= 4\sqrt a \mathop \smallint \limits_0^a \sqrt x \;dx\)

\(= 4\sqrt a \left[ {\frac{{{x^{\frac{3}{2}}}}}{{\frac{3}{2}}}} \right]_0^a\)

\(= \frac{8}{3}\sqrt a \left[ {{a^{\frac{3}{2}}} - 0} \right]\)

\(Area= \frac{8}{3}{a^2}\)