The area bounded by the parabola {y^2} = 4ax | vedclass

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(C) The equation of the parabola is ${y^2} = 4ax$. The latus rectum is the line $x = a$.To find the area bounded by the parabola and its latus rectum,

Vedclass · Accepted Answer

$\frac{8}{3}{a^2} 	ext{ sq. unit}$

Vedclass · Answer

(C) The equation of the parabola is ${y^2} = 4ax$. The latus rectum is the line $x = a$.To find the area bounded by the parabola and its latus rectum,we integrate with respect to $x$ from $0$ to $a$.Since the parabola is symmetric about the $x$-axis,the total area is twice the area in the first quadrant.$	ext{Area} = 2 \int_0^a y \, dx = 2 \int_0^a \sqrt{4ax} \, dx$$= 2 	imes 2\sqrt{a} \int_0^a x^{1/2} \, dx$$= 4\sqrt{a} \left[ \frac{x^{3/2}}{3/2} ight]_0^a$$= 4\sqrt{a} 	imes \frac{2}{3} \left[ a^{3/2} - 0 ight]$$= \frac{8}{3} \sqrt{a} 	imes a\sqrt{a} = \frac{8}{3} a^2 	ext{ sq. unit}$

The area bounded by the parabola ${y^2} = 4ax$ and its latus rectum is:

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