Power Spectral Density of Finite Signals
Assuming the systems are sampled and we are looking for the discrete power spectral density, meaning a relative indication of the power level of signal versus frequency, and as suggested in the comments, the sampling rate is an integer multiple of the 1 Hz period, then each case can be described as follows:

For case 1, the Discrete Fourier Transform, if scaled by N, would result in the coefficients of Euler's Identity for a sine wave (this is explained simply by realizing that the Fourier Transform is the correlation to each exponential frequency term $e^{j\omega t}$ and as Euler's identity illuminates, the sine wave consists of positive and negative exponential frequency components. 

$$sin(2\pi t) = -0.5j e^{2\pi t} + 0.5j e^{2\pi t} $$

The each of the two bins that have a non-zero magnitude ($\pm 1$ Hz) would have a complex value of +/-0.5j.

The power for each bin is found by complex conjugate multiplication or 0.25 in each case.

If the sampling rate is NOT an integer multiple of the frequency present (1 Hz in this case), spectral leakage would result such that the power would be distributed across all the other bins. The nature of this distribution is dependent on the actual sampling rate used.

For case 2 We are now multiplying our original signal in time by a smaller rectangular window. If you take the DFT of that window (all zeros from 0 to &lt;2, ones from 2 to 5 and all zeros from >5 to 10, with the samples dictated by the sampling rate used) the result will be an aliased Sinc function; the specific result will depend on the actual sampling rate used. However since multiplication in time is equivalent to convolution in frequency, this specific aliased Sinc function result would convolve in frequency with the result from case 1 (as a circular convolution) to the provide the modified power spectral density.