Question

The area of the plane figure bounded by one leaf of the curve r=4cos4φ is equal to k⋅

Answer 1

The area of one leaf of the curve r=4cos4φ can be calculated using polar coordinates, integrating 1/2 times the squared radius function over the interval 0 to π/4.

Step-by-step explanation:

The calculation of the area of one leaf of the curve given by the polar equation r = 4cos4φ can be found using an integral. In polar coordinates, the area of a sector of a curve is given by the integral
`A = 1/2 ∫ (r^2) dφ`. In this case, because there are multiple leaves for the function r = 4cos4φ, and we are interested in only one leaf, we would integrate over the interval that corresponds to that single leaf. Assuming the student is referring to the area within one full oscillation, φ would range from 0 to π/4. The integral becomes
`A = 1/2 ∫_(0)^(pi/4) (4cos4φ)^2 dφ`. The calculation would then yield the area k, which represents the area of one leaf of the given polar curve.