Question:
Consider the functions $f, g: R \rightarrow R$ defined by $f(x)=x^2+\frac{5}{12} \text { and } g(x)= \begin{cases} 2\left(1-\frac{4|x|}{3}\right), & |x| \leq \frac{3}{4}, \\0, & |x|>\frac{3}{4} \end{cases}$ If $\alpha$ is the area of the region $\left\{( x , y ) \in R \times R :| x | \leq \frac{3}{4}, 0 \leq y \leq \min \{f( x ), g( x )\}\right\},$ then the value of $9 \alpha$ is _____
Consider the functions $f, g: R \rightarrow R$ defined by $f(x)=x^2+\frac{5}{12} \text { and } g(x)= \begin{cases} 2\left(1-\frac{4|x|}{3}\right), & |x| \leq \frac{3}{4}, \\0, & |x|>\frac{3}{4} \end{cases}$ If $\alpha$ is the area of the region $\left\{( x , y ) \in R \times R :| x | \leq \frac{3}{4}, 0 \leq y \leq \min \{f( x ), g( x )\}\right\},$ then the value of $9 \alpha$ is _____
Updated On: Feb 8, 2024
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Correct Answer: 6
Solution and Explanation
The Value of \(9\alpha \;\text{is}\; \underline{6}.\)
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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