In this video I explain how to find the area of a four leaf rose petal bounded by a polar curve using double integration. I use the polar curve r=2Sin2Θ to find the area of a single petal and then explain how to scale this up to find the area of all 4 petals.
I begin this tutorial using a graph plotting tool to understand how rose petals are created using polar curves. I use a generic example r = nSin(kΘ) to help understand the parameters n and k and explain how the polar curve is affected as these parameters are varied.
I then explain how the area is calculated using a double integral which effectively sums infinitesimally small slices of area to give the area of the whole region and how an infinitesimally small slice of area is give by rdrdΘ.
I then show how to find the integration limits of the inner integral which is in the r direction.
For the outer integral, as Θ rotates about the origin, I use a graph plotting tool to demonstrate how one half of a petal is plotted and that it begins at r=0 and ends at r=2. Using this information I demonstrate that the area of one petal can be found by integrating between the angles of Θ = 0 and Θ = π/2 and explain how the whole area can be calculated by multiplying the area of a single petal by 4.
I finish the video by evaluating the inner integral first and I then move on to evaluating the outer integral with the help of trig identities.