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^{2}d (theta).

this can be interpreted as δA= ∏r

^{2}δ(theta)/2∏ (treating the area element as the area of a sector of a circle with angle δ(theta).)

taking limit of δ(theta)→0,

dA= [STRIKE]∏[/STRIKE]r

^{2}d(theta)/2[STRIKE]∏[/STRIKE]=1/2 (r

^{2}d(theta) )

there fore A=1/2∫r

^{2}d(theta).

By the same logic, shouldn't length of a polar curve be L=∫[STRIKE]2∏[/STRIKE]r d(theta)/[STRIKE]2∏[/STRIKE]=∫r d(theta)?

the actual equation for curve length is L=∫√(r

^{2}+r'

^{2})d(theta).

why is this approach giving me an incorrect equation in the second case but a correct one in the first?

ps. the equation for length of a curve can also be derived by partially differentiating the equation for area by r: ∂A/∂r=L. This is again consistent with looking at length of a curve as the rate of change of area.