The area of the region bounded by y^{2}=8x an | vedclass

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

Question

(C) To find the area of the region bounded by the parabolas $y^{2}=8x$ and $y^{2}=16(3-x)$,we first find their points of intersection.Equating the two

Vedclass · Accepted Answer

$16$

Vedclass · Answer

(C) To find the area of the region bounded by the parabolas $y^{2}=8x$ and $y^{2}=16(3-x)$,we first find their points of intersection.Equating the two expressions for $y^{2}$:$8x = 16(3-x)$$8x = 48 - 16x$$24x = 48$$x = 2$Substituting $x=2$ into $y^{2}=8x$,we get $y^{2}=16$,so $y = \pm 4$.The region is symmetric about the $x$-axis. The area can be calculated by integrating with respect to $y$ from $-4$ to $4$:$	ext{Area} = \int_{-4}^{4} (x_{R} - x_{L}) dy$From $y^{2}=8x$,$x_{L} = \frac{y^{2}}{8}$.From $y^{2}=16(3-x)$,$x_{R} = 3 - \frac{y^{2}}{16}$.$	ext{Area} = \int_{-4}^{4} \left(3 - \frac{y^{2}}{16} - \frac{y^{2}}{8}ight) dy = 2 \int_{0}^{4} \left(3 - \frac{3y^{2}}{16}ight) dy$$= 2 \left[ 3y - \frac{3y^{3}}{16 	imes 3} ight]_{0}^{4} = 2 \left[ 3y - \frac{y^{3}}{16} ight]_{0}^{4}$$= 2 \left( 3(4) - \frac{4^{3}}{16} ight) = 2 (12 - 4) = 2(8) = 16$.

The area of the region bounded by $y^{2}=8x$ and $y^{2}=16(3-x)$ is equal to

Similar Questions

If the area of the region $\{(x, y): 1+x^2 \leq y \leq \min \{x+7, 11-3x\}\}$ is $A$,then $3A$ is equal to

The area of the region bounded by the curve $y = \max \{| x |, x | x - 2 |\}$,the $x$-axis,and the lines $x = -2$ and $x = 4$ is equal to . . . . . . .

The figure shows a portion of the graph $y=2x-4x^3$. The line $y=c$ is such that the areas of the regions marked $I$ and $II$ are equal. If $a$ and $b$ are the $x$-coordinates of $A$ and $B$ respectively,then $a+b$ equals:

The area bounded by the curves $y^2 - x = 0$ and $y - x^2 = 0$ is

The area (in sq. units) of the region bounded by the parabola $y = x^2 + 2$ and the lines $y = x + 1$,$x = 0$,and $x = 3$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module