integral of 0dt=?

where is an arbitrary constant.
You need to understand the distinction between definite integrals (limits prescribed) and indefinite integral (no limits). A definite integral of 0 will be 0, since the constant of integration (from the indefinite integral) cancels.
When you talk about area under a curve, you are talking about definite integrals.
Answer: If your equation is y=0, and you want to find the area under the curve, then you can immediately jump to the conclusion that for all x, there is no area under that curve.
If you then decide y=0 is really a slope, its the slope of dZ/dx, "what is the shape of a line when its dZ/dx = 0 " .
Then the line is a horizontal, Z=C
Now what if decide that dZ/dx really was a line ? What about the area under it ? Well there is a value of C for all x, but the different between any two x's is 0 , so despite there being values for the end points, the area is 0.
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