The figure below shows the graph of r = 1 + cosθ. Use...

The figure below shows the graph of $r = 1 + \cos \theta$. Use $\frac{1}{2} \int_{\theta=a}^{\theta=b} (f(\theta))^2 d\theta$ to find the area of the shaded region. Notes: 1. $\int \cos^2 u \, du = \frac{u}{2} + \frac{1}{4} \sin(2u) + C$

Transcript

00:01 So, we are given a graph in that for the graph the equation is r is equals to 2 plus sine of theta.

00:12 So, you have to find the area by using the rebel integration.

00:16 So, required area this should be equals to the integration.

00:22 The first reading is for theta which will vary from 0 to pi by 2 for the quarter half and similarly second reading is for r which will vary from 0 to 2 plus sine of theta and here it will be r dr d theta that we are considering in a small arc that will be represented something like this in between that whole of that arc.

00:50 So, just like this we are going to find it out.

00:53 So, this will be goes to.

00:55 So, let us take 2 as a common.

00:57 So, here it will be 2 common integration sorry do one correction we just said theta should vary from minus pi by 2 to plus pi by 2.

01:09 So, theta will be as it is minus pi by 2 to plus pi by 2 here it will becomes integral of r with respect to r is r square by 2 and the limits will go from 0 to 2 plus sine theta and now we have to integrate with respect to d theta.

01:27 So, 2 will be taken as constant.

01:29 So, 2 by 2 will become here and minus pi by 2 to plus pi by 2 here it will be 2 plus sine of theta that whole square upstairs substituting the limit and here it should be d theta.

01:43 So, 2 2 get cancelled out.

01:45 Now this will become minus pi by 2 to plus pi by 2 opening the bracket by using a plus b formula...

Question

The figure below shows the graph of $r = 1 + \cos \theta$. Use $\frac{1}{2} \int_{\theta=a}^{\theta=b} (f(\theta))^2 d\theta$ to find the area of the shaded region. Notes: 1. $\int \cos^2 u \, du = \frac{u}{2} + \frac{1}{4} \sin(2u) + C$

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