Transcription of Net Change and Displacement - HWS Department of ...
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Net Change and Displacement - HWS Department of ...
Math 131,applicationsmotion:velocity and net Change 1 Net Change and DisplacementWe have seen that the definite integral baf(x)dxmeasures thenet areaunderthe curvey=f(x)on the interval[a,b]. Any part of the region below thex-axiscontributes negatively-signed area to this net calculation. To find thetotal areaenclosed byfon[a,b], one needs to evaluate the definite integral of the absolutevalue off(x): ba|f(x)|dxWe can apply this idea to other contexts. In Calculus I we interpreted the firstand second derivatives as velocity and acceleration in the context of motion. Inparticular, l s(t) =position at timet. s (t) =v(t) =velocity at timet. s (t) =v (t) =a(t) =acceleration at our point of view and using antiderivatives, since positions(t)is anantiderivative of velocttyv(t), by the FTC (II) we have bav(t)dt=s(t) ba=s(b) s(a).Buts(b) s(a)represents thenet distance traveledordisplacementof the object dur-ing the time interval because it is the difference in locations at the end and start ofthe time we wanted thetotal distance traveledrather than the net distance traveled, justas with the area problem, we would compute the integral of|v(t)|instead.
Net Change and Displacement We have seen that the definite integral Zb a f(x)dx measures the net area under the curve y = f(x) on the interval [a,b]. Any part of the region below the x-axis ... ing along a straight line at time t is denoted by s(t) relative to the origing and that its ve-locity is denoted by v(t). Then 1.
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