A student claims that the area between two polar curves r=f(θ) and r=g(θ) on α≤θ≤β can be found by integrating ∫αβ(f(θ)−g(θ))dθ. Another student disagrees and asserts that squaring the radii is essential.
Address the dispute by responding to the parts below.
a. For a single polar curve r=h(θ), explain—using geometric reasoning—why the differential element of area for a small angle dθ is proportional to h(θ)2dθ rather than to h(θ)dθ.
b. Using your reasoning from part (a), derive an expression (in integral form) that correctly represents the area between two polar curves on α≤θ≤β. Do not simplify the expression.
c. Provide a specific example of two curves on a closed interval of θ where integrating the difference of the radii (without squaring) gives a numerical result different from the correct area. Compute both the incorrect and correct results, and explain the discrepancy.