Mathematics

Mathematics

Problem

For a plane curve defined parametrically by x=f(t)x=f(t) and y=g(t)y=g(t) on the interval a≤t≤ba \le t \le b, which of the following definite integrals would compute the length of the curve? a) ∫ab([f′(t)]2+[g′(t)]2) dt\displaystyle \int_{a}^{b}\left(\bigl[f'(t)\bigr]^{2}+\bigl[g'(t)\bigr]^{2}\right)\,dt b) ∫ab[f′(t)]2+[g′(t)]2 dt\displaystyle \int_{a}^{b}\sqrt{\bigl[f'(t)\bigr]^{2}+\bigl[g'(t)\bigr]^{2}}\,dt c) ∫abf(t)2+g(t)2 dt\displaystyle \int_{a}^{b}\sqrt{f(t)^{2}+g(t)^{2}}\,dt d) ∫ab(∣f′(t)∣+∣g′(t)∣) dt\displaystyle \int_{a}^{b}\bigl(|f'(t)|+|g'(t)|\bigr)\,dt

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