If I have a graph of Acceleration against time. Can I integrate this curve in order to find velocity and displacement?
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Yes. $a(t) = dv(t)/dt$ and $v(t) = dx(t)/dt$, where $a$ is acceleration, $v$ is velocity and $x$ is position (in one dimension: all this works generally of course). So integrating once gives you velocity, twice gives you displacement. This is how inertial navigation works, by the way: the device measures acceleration (which is all it can measure locally) and integrates twice to get position.
(As noted by Perfi in the comment below, you also need the initial conditions: if you know only acceleration as a function of time you need both the velocity and position at some point (usually the start). These two initial conditions correspond to the constants of integration.)
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2$\begingroup$ Note that you also need the initial conditions; integrating acceleration with respect to time will only give you the change in velocity from some initial value. Same goes for position, as integrating the acceleration will give you only the displacement resulting from that acceleration, and not from any initial constant velocity movement the body. $\endgroup$ Commented Mar 6, 2017 at 12:28
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$\begingroup$ @Perfi: thanks, I've added that. Also thanks for fixing the idiot typo. $\endgroup$– user107153Commented Mar 6, 2017 at 13:17