The area of the region S = {(x, y) : y^{2} le | vedclass

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

Question

(B) The region $S$ is bounded by the parabola $y^{2} = 8x$ and the line $y = \sqrt{2}x$ for $x \geq 1$.First,find the intersection points of $y^{2} =

Vedclass · Accepted Answer

$\frac{11 \sqrt{2}}{6}$

Vedclass · Answer

(B) The region $S$ is bounded by the parabola $y^{2} = 8x$ and the line $y = \sqrt{2}x$ for $x \geq 1$.First,find the intersection points of $y^{2} = 8x$ and $y = \sqrt{2}x$:$(\sqrt{2}x)^{2} = 8x \Rightarrow 2x^{2} = 8x \Rightarrow 2x(x - 4) = 0$.Thus,the curves intersect at $x = 0$ and $x = 4$.Since the region is defined for $x \geq 1$,the limits of integration are from $x = 1$ to $x = 4$.The area $A$ is given by:$A = \int_{1}^{4} (\sqrt{8x} - \sqrt{2}x) \, dx$$A = \int_{1}^{4} (2\sqrt{2}\sqrt{x} - \sqrt{2}x) \, dx$$A = 2\sqrt{2} \left[ \frac{x^{3/2}}{3/2} \right]_{1}^{4} - \sqrt{2} \left[ \frac{x^{2}}{2} \right]_{1}^{4}$$A = \frac{4\sqrt{2}}{3} (4^{3/2} - 1^{3/2}) - \frac{\sqrt{2}}{2} (4^{2} - 1^{2})$$A = \frac{4\sqrt{2}}{3} (8 - 1) - \frac{\sqrt{2}}{2} (16 - 1)$$A = \frac{4\sqrt{2}}{3} (7) - \frac{\sqrt{2}}{2} (15)$$A = \frac{28\sqrt{2}}{3} - \frac{15\sqrt{2}}{2} = \frac{56\sqrt{2} - 45\sqrt{2}}{6} = \frac{11\sqrt{2}}{6}$.

The area of the region $S = \{(x, y) : y^{2} \leq 8x, y \geq \sqrt{2}x, x \geq 1\}$ is

Similar Questions

For $x \in [0, 2\pi]$,the area of the region bounded by the curves $y = x + \sin x$ and $y = x$ is:

The area of the region described by $\{(x, y) \mid x^2+y^2 \leq 1\}$ and $\{y^2 \leq 1-x\}$ is

The area (in sq. units) bounded by the curves $x^2=9y$,$(x-6)^2=9y$ and the $X$-axis is

If $(\alpha, \beta)$ is the stationary point of the curve $y=2x-x^2$,then the area bounded by the curves $y=2^x, y=2x-x^2, x=0$ and $x=\alpha$ is

If $f(x)$ is a continuous,increasing,and odd function such that $\int_{-1}^{4} f(x) \,dx = 10$ and $\int_{0}^{1} f(x) \,dx = \frac{3}{2}$,then the area bounded by $y = f(x)$,the $x$-axis,and the ordinates $x = -4$ and $x = 4$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module