Q. 8. - Calculus

Question

Consider the three-petaled polar rose defined by $r=\cos 3\theta$ .Explain why the definite integral $\frac{1}{2}{\int }_0^{2\pi }{\cos }^{2}3\theta d\theta$calculates twice the area bounded by the petals of this rose.

Step-by-Step Solution

Answer

If the area is calculated in the interval $[0,2\pi ]$t gives twice the area bounded by the petals of the polar curve $r=\cos 3\theta$

Step 1: Given information

The definite integral is $\frac{1}{2}{\int }_0^{2\pi }{\cos }^{2}3\theta d\theta$

Consider the polar curve $r=\cos 3\theta$

Step 2: The objective is to give the reason why the area of the curve 1 2 ∫ 0 2 π cos 2 3 θ d θ   is twice the actual area

The curve $r=\cos 3\theta$traced twice in the interval $[0,2\pi ]$

As a result, calculating the area in the interval $\left[0,2\pi \right]$

It delivers twice the area enclosed by the polar curve's  $r=\cos 3\theta$ petals.