Question:
The area of the region given by $\left\{(x, y): x y \leq 8,1 \leq y \leq x^2\right\}$ is :
The area of the region given by $\left\{(x, y): x y \leq 8,1 \leq y \leq x^2\right\}$ is :
Updated On: Sep 24, 2024
- $16 \log _e 2-\frac{14}{3}$
- $8 \log _e 2-\frac{13}{3}$
- $8 \log _e 2+\frac{7}{6}$
- $16 \log _e 2+\frac{7}{3}$
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The Correct Option is A
Solution and Explanation
Area =1∫2(x2−1)dx+2∫8(x8−1)dx
=(3x3)12+8(ℓnx)28−(x)18
=37+8(2ℓn2)−7
=16ln2−314
So, the correct option is (A): 16loge 2−314
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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