Why is the definite integral described as 'accumulation' — what does that mean physically?
A definite integral adds up a rate over an interval to give the total change.
A definite integral adds up a rate over an interval to give the total change. If v(t) is velocity, then ∫ v(t) dt over a time interval is the total displacement — you're accumulating all the little distances v·dt. The same idea turns a flow rate into total volume, or a marginal cost into total cost. So integration is summation of a continuously changing quantity, not just 'area for its own sake'.
Key point
∫ (rate) dt = total accumulated change (e.g. ∫ velocity dt = displacement).
Common mistake
thinking integrals only compute abstract geometric area.
Memory tip
integrate a rate to accumulate the total — velocity → distance.
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