Consider the integral C F(r) d r , where F = [xy, -y 2 ].
Question:
Consider the integral ∫C F(r) • dr, where F = [xy, -y2].
(a) Find the value of the integral when r = [cos t, sin t], 0 ≤ t ≤ π/2. Show that the value remains the same if you set t = -p or t = p2 or apply two other parametric transformations of your own choice.
(b) Evaluate the integral when C: y = xn, thus r = [t, tn], 0 ≤ t ≤ 1 where n = 1, 2, 3, · · ·. Note that these infinitely many paths have the same endpoints.
(c) What is the limit in (b) as n → ∞? Can you confirm your result by direct integration without referring to (b)?
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a r cos t sin t r sin t cos t From F xy y 2 we obtain Fr t cos t sin t sin 2 t Hence the integra...View the full answer
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