Question:
The area of the region \[\left\{ (x, y) : y^2 \leq 4x, \, x<4, \, \frac{xy(x - 1)(x - 2)}{(x - 3)(x - 4)}>0, \, x \neq 3 \right\}\]is
The area of the region \[\left\{ (x, y) : y^2 \leq 4x, \, x<4, \, \frac{xy(x - 1)(x - 2)}{(x - 3)(x - 4)}>0, \, x \neq 3 \right\}\]is
Updated On: Mar 20, 2025
- \( \frac{16}{3} \)
- \( \frac{64}{3} \)
- \( \frac{8}{3} \)
- \( \frac{32}{3} \)
Hide Solution
Verified By Collegedunia
The Correct Option is D
Solution and Explanation
\[ y^2 \leq 4x, \quad x < 4 \]
\[ \frac{xy(x-1)(x-2)}{(x-3)(x-4)} > 0 \]
Case - I: \( y > 0 \)
\[ \frac{x(x-1)(x-2)}{(x-3)(x-4)} > 0, \quad x \in (0,1) \cup (2,3) \]
Case - II: \( y < 0 \)
\[ \frac{x(x-1)(x-2)}{(x-3)(x-4)} < 0, \quad x \in (1,2) \cup (3,4) \]
Area:
\[ \text{Area} = 2 \int_{0}^{4} \sqrt{x} \, dx \]
\[ = 2 \cdot \frac{2}{3} \left[ x^{3/2} \right]_{0}^{4} = \frac{32}{3} \]
Was this answer helpful?
0
0
Top Questions on integral
- Evaluate the integral: $$ \int_0^{\pi/4} \frac{\ln(1 + \tan x)}{\cos x \sin x} \, dx $$
- Find the value of the integral: \[ \int_0^\pi \sin^2(x) \, dx. \]
- Evaluate the integral \( \int_0^1 \frac{\ln(1 + x)}{1 + x^2} \, dx \)
- Evaluate the integral: $$ \int_0^1 \frac{\ln (1+x)}{1+x^2} \, dx $$
- Evaluate $ \int_{0}^{1} (3x^2 + 2x) \, dx $.
View More Questions
Questions Asked in JEE Main exam
A force \( \vec{f} = x^2 \hat{i} + y \hat{j} + y^2 \hat{k} \) acts on a particle in a plane \( x + y = 10 \). The work done by this force during a displacement from \( (0,0) \) to \( (4m, 2m) \) is Joules (round off to the nearest integer).
- JEE Main - 2025
- Work and Energy
- Consider a long thin conducting wire carrying a uniform current \( I \). A particle having mass \( M \) and charge \( q \) is released at a distance \( a \) from the wire with a speed \( v_0 \) along the direction of current in the wire. The particle gets attracted to the wire due to magnetic force. The particle turns round when it is at distance \( x \) from the wire. The value of \( x \) is:
- JEE Main - 2025
- Magnetic Field
- Two equal sides of an isosceles triangle are along \( -x + 2y = 4 \) and \( x + y = 4 \). If \( m \) is the slope of its third side, then the sum of all possible distinct values of \( m \) is:
- JEE Main - 2025
- 3D Geometry
- Let the position vectors of the vertices A, B, and C of a tetrahedron ABCD be \( \hat{i} + 2\hat{j} + \hat{k} \), \( \hat{i} + 3\hat{j} - 2\hat{k} \), and \( 2\hat{i} + \hat{j} - \hat{k} \) respectively. The altitude from the vertex D to the opposite face ABC meets the median line segment through A of the triangle ABC at the point E. If the length of AD is \( \frac{\sqrt{10}}{3} \) and the volume of the tetrahedron is \( \frac{\sqrt{805}}{6\sqrt{2}} \), then the position vector of E is:
- JEE Main - 2025
- Geometry and Vectors
- If \( x_1, x_2, x_3, x_4 \) are in GP (Geometric Progression), then we subtract 2, 4, 7, and 8 from \( x_1, x_2, x_3, x_4 \) respectively, then the resultant numbers are in AP (Arithmetic Progression). Then the value of \( \frac{1}{24} (x_1 \cdot x_2 \cdot x_3 \cdot x_4) \) is:
- JEE Main - 2025
- Arithmetic Progression
View More Questions