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Different magnitude values when computing f-domain function by hand and with fft matlab command.

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Hello everyone, I have the following function in time-domain and its correspondent f-domain transform:
With t1 as [0s,8s) and f1 as [-50hz,50hz)
When I plot, I get the following:
Now, I tried using fft command to the time-domain function and see what I got, even if I choose the same t1 and f1 vectors. For my surprise, we have differences when plotting:
Why's that? Is it something about fft command?
Here's how I'm placing the formulas in case you think there's a mistake in there:
Can you help me find a reason why is this happening?

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