The area (in square units) of the region boun | vedclass

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

Question

(B) Given equations are $y^2=4(x-2)$ and $y=2x-8$. From the line equation,$2x = y+8$,so $x = \frac{y+8}{2} = \frac{y}{2} + 4$. Substituting $x$ in the

Vedclass · Accepted Answer

$9$

Vedclass · Answer

(B) Given equations are $y^2=4(x-2)$ and $y=2x-8$. From the line equation,$2x = y+8$,so $x = \frac{y+8}{2} = \frac{y}{2} + 4$. Substituting $x$ in the parabola equation: $y^2 = 4(\frac{y}{2} + 4 - 2) = 4(\frac{y}{2} + 2) = 2y + 8$. $y^2 - 2y - 8 = 0 \implies (y-4)(y+2) = 0$. Thus,the intersection points are $y=4$ and $y=-2$. The area $A$ is given by $\int_{-2}^{4} (x_{line} - x_{parabola}) dy$. $x_{line} = \frac{y+8}{2}$ and $x_{parabola} = \frac{y^2}{4} + 2$. $A = \int_{-2}^{4} (\frac{y+8}{2} - (\frac{y^2}{4} + 2)) dy = \int_{-2}^{4} (\frac{y}{2} + 4 - \frac{y^2}{4} - 2) dy = \int_{-2}^{4} (\frac{y}{2} + 2 - \frac{y^2}{4}) dy$. $A = [\frac{y^2}{4} + 2y - \frac{y^3}{12}]_{-2}^{4}$. $A = (\frac{16}{4} + 8 - \frac{64}{12}) - (\frac{4}{4} - 4 - \frac{-8}{12}) = (4 + 8 - \frac{16}{3}) - (1 - 4 + \frac{2}{3}) = (12 - \frac{16}{3}) - (-3 + \frac{2}{3}) = \frac{20}{3} - (-\frac{7}{3}) = \frac{27}{3} = 9$ square units.

The area (in square units) of the region bounded by the parabola $y^2=4(x-2)$ and the line $y=2x-8$ is:

Similar Questions

The area of the region $R=\{(x,y):xy \le 8, 1 \le y \le x^{2}, x \ge 0\}$ is

The area bounded by the curves $y=x^2$ and $y-6=-|x|$ is

Let $f: R \to R$ be a function such that $f(x) + 3f(\frac{\pi}{2} - x) = \sin x, x \in R$. Let the maximum value of $f$ on $R$ be $\alpha$. If the area of the region bounded by the curves $g(x) = x^2$ and $h(x) = \beta x^3, \beta > 0$, is $\alpha^2$, then $30\beta^3$ is equal to ————

The area (in $sq. units$) of the region $A = \{(x,y) : \frac{y^2}{2} \le x \le y + 4\}$ is

The area (in sq. units) of the region described by $A = \{(x, y) : x^2 + y^2 \leq 1 - x\}$ is

Vedclass Test Series

Exam Paper Generator

Online Exam Module