Final answer:
After calculating the definite integral for the given polar curve r=7e⁰.⁶θ from 0 to 1/4, the area doesn't match any of the provided options, suggesting that there may be a typo or error in the given choices.
Step-by-step explanation:
The area enclosed by the polar curve r=7e⁰.⁶θ over the interval 0 < θ < 1/4 is calculated using the integral expression for the area in polar coordinates, which is A = 1/2 ∫ θ_2 θ_1 r(θ)^2 dθ. Applying the given limits of integration from 0 to 1/4 to the given function, we have:
A = 1/2 ∫^1/4_0 (7e⁰.⁶θ)^2 dθ = 49/2 ∫^1/4_0 e¹.2θdθ = 49/2 [e¹.2θ/1.2] from 0 to 1/4 = (49/(2×1.2))(e¹.2(1/4)-1)
Calculating this definite integral we get:
A = 49/(2×1.2)(e°.3 - 1)
Since none of the given answer choices match the calculation, there is a possibility that the question contains a typo, or the given choices are incorrect. The student is advised to double-check the question and the answer choices.