Question:
The area (in s units) of the region $\{ (x , y) : x \geq 0 , x + y \leq 3, x^2 \leq 4 y$ and $y \leq 1 + \sqrt{x} \}$ is
The area (in s units) of the region $\{ (x , y) : x \geq 0 , x + y \leq 3, x^2 \leq 4 y$ and $y \leq 1 + \sqrt{x} \}$ is
Updated On: Feb 14, 2025
- $\frac{3}{2}$
- $\frac{7}{3}$
- $\frac{5}{2}$
- $\frac{59}{12}$
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The Correct Option is C
Solution and Explanation
Area of shaded region
$= \int\limits^{1}_{0}$$\left(\sqrt{x}+1-\frac{x^{2}}{4}\right)dx+ $$= \int\limits^{2}_{1}$$\left(\left(3-x\right)-\frac{x^{2}}{4}\right)dx$
$= \frac{5}{2 }$ s unit
$= \int\limits^{1}_{0}$$\left(\sqrt{x}+1-\frac{x^{2}}{4}\right)dx+ $$= \int\limits^{2}_{1}$$\left(\left(3-x\right)-\frac{x^{2}}{4}\right)dx$
$= \frac{5}{2 }$ s unit
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Concepts Used:
Applications of Integrals
There are distinct applications of integrals, out of which some are as follows:
In Maths
Integrals are used to find:
- The center of mass (centroid) of an area having curved sides
- The area between two curves and the area under a curve
- The curve's average value
In Physics
Integrals are used to find:
- Centre of gravity
- Mass and momentum of inertia of vehicles, satellites, and a tower
- The center of mass
- The velocity and the trajectory of a satellite at the time of placing it in orbit
- Thrust