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So for the first question:

If this is sampled at 10kHz, then the amplitude is scaled by 10000. In the DTFT, the frequency 3.5kHz gets mapped to 3.5/10* 2pi=0.7pi. So this point lies outside the range of the integration.

The next non-zero point is, 10-3.5=6.5kHz. This gets mapped to 6.5/10*2pi=1.3pi. This also lies outside of the integration range.

So the integral in the range 0.75pi to pi contains only zero points, which means the integral is zero. Is this correct?

For the second question,

The frequency 3kHz gets mapped to 3/8*2pi=0.75pi. This is the lower limit of the integral. 3.5kHz gets mapped to 3.5/8*2pi=0.875pi. So the spike at 3.5kHz is inside the integral. The next non zero point, 8-3.5=4.5kHz gets mapped to 1.125pi, which is outside the integral.

So in this case, the integral from 0.75pi to pi contains only one non-zero spike at 0.875pi of amplitude 8000. However, I don't know what is the integral of a spike. Please help.

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