Final answer:
To integrate f over the given curve x³/y, you can parameterize the curve using x as the parameter. By using x = t and y = t³, where -∞ < t < ∞, you can then express f in terms of t as f(x, y) = t². Finally, integrate f(t²) with respect to t over the range -∞ to ∞ to obtain the result ∞.
Step-by-step explanation:
To integrate f over the given curve x³/y, we first need to parameterize the curve. Let's use x as the parameter. We can rewrite the curve as x = t and y = t³, where -∞ < t < ∞. Now, we can express f in terms of t as f(x, y) = t³/t = t². To integrate over the curve, we evaluate the integral of f with respect to t over the range -∞ to ∞.
The integral becomes: ∫(t²) dt with limits from -∞ to ∞. Evaluating this integral, we get (t³/3) with limits from -∞ to ∞. Substituting the limits, the final result is (∞³/3) - (-∞³/3), which is 2∞³/3 or simply ∞.