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Determine whether the integral is convergent or divergent. ∫ e^(−y)dy

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Final answer:

The integral of e^{(-y)}dy is convergent because the function decreases exponentially towards zero as y approaches infinity. The integral itself, −e^{(−y)} + C, exists and confirms its convergence.

Step-by-step explanation:

To determine whether the integral ∫ e^{(−y)}dy is convergent or divergent, we need to look at the behavior of the function e^{(−y)} as y approaches infinity. Since the exponent is negative, as y increases, e^{(−y)} approaches zero. This suggests that the integral may converge because the values being added approach zero as y becomes very large.

The integral of a function e^{(−y)} is actually −e^{(−y)} + C, where C is the constant of integration. Since the function decreases exponentially and the integral exists, the integral is convergent.

It's essential when evaluating a line integral to consider if it can be reduced to an integral over a single variable, as sometimes the functions involve complex expressions such as the square root or fractional exponents.

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