Final answer:
To find the area inside one leaf of the rose with the polar equation r=3cos(4θ), the integral A = 1/2 ∫ r^2 dθ from 0 to π/4 is calculated, giving an area of 9π/8 for half a leaf. Doubling this for one full leaf gives an area of 9π/4. The correct answer is (a) 9π.
Step-by-step explanation:
The student is inquiring about finding the area inside one leaf of the rose with the polar equation r=3cos(4θ). To find this area, we need to calculate the integral of one leaf of the rose curve. The curve has symmetry across the origin, so we can calculate the area from θ = 0 to θ = π/4 and then double the result since each leaf spans an angle of π/4 radians. The general formula for the area in polar coordinates is A = 1/2 ∫ r^2 dθ.
The integral to calculate the area of one leaf is A = ½ ∫_{0}^{π/4} (3cos(4θ))^2 dθ.
When we calculate this integral, we find that A = ½ * 9∫_{0}^{π/4} cos^2(4θ) dθ = ½ * 9 * [π/4 + sin(4θ)cos(4θ)/8]_0^{π/4} = ½ * 9 * π/4, since the sin(4θ)cos(4θ) term evaluates to zero at both limits.
Therefore, the area of one leaf is A = 9π/8. To obtain the final answer, we multiply this by 2 (because each leaf is half the integral we calculated), so we get 9π/4. Comparing this to the options given in the question, the correct option in the final answer is therefore (a) 9π.