Question:

Area of loop of the curve $r = a \sin \, 2 \theta$ is

Updated On: Jul 6, 2022
  • $\frac{\pi a^2}{8}$
  • $\frac{\pi a}{8}$
  • $\frac{ 3 \pi a^2}{8}$
  • $\frac{ 7\pi a^2}{8}$
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The Correct Option is C

Solution and Explanation

Given equation is equation of a curve in polar form is $r = a \sin 2 \theta$ $ r = 0 \Rightarrow \theta = \frac{\pi}{2}$ similarly, $\theta = 0 , \frac{\pi}{2}, \pi, , \frac{3\pi}{2} , 2 \pi$ Required area = Area of one loof $ = \int{\pi/2}_0 \frac{1}{2} r^2 d \theta$
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Top Questions on applications of integrals

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Notes on applications of integrals

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