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

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(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:

Similar Questions

The area bounded by the curves $y = \cos x$ and $y = \sin x$ and the ordinates $x = 0$ and $x = \frac{\pi}{4}$ is:

If the area bounded by the curve $x^2=4y$,the $X$-axis,and the line $x=4$ is divided into two equal areas by the line $x=\alpha$,then the value of $\alpha$ is ...

The area of the region bounded by the line $y=2x+1$,the $X$-axis,and the ordinates $x=-1$ and $x=1$ is

The area of the region $\{(x, y): 0 \leq x \leq \frac{9}{4}, 0 \leq y \leq 1, x \geq 3y, x+y \geq 2\}$ is

If the area bounded by the curve $x^2 = by$ and the lines $y = 1, y = 4$ in the first quadrant is $28$ sq. units, then the value of $b$ is...

Vedclass Test Series

Exam Paper Generator

Online Exam Module