从变化率到累积的变化

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Water Those
Moor Xu

From the Fundamental Theorem of Calculus, we have that $\int_{a}^{b} f(x) dx = F(b) - F(a)$ , where $F(x)$ is the antiderivative of $f(x)$ . In other words, the definite integral of a function in a certain region is the amount of change of its antiderivative. The usefulness of this becomes apparent when we encounter a problem such as the following:

Bob's pool is filling with water at a rate given by the function $R(t) = 3t^2 - 26t + 84$ , where $t$ is the time in hours that has elapsed and $R(t)$ is in gallons per hour. What is the net change in volume (in gallons) of water in the pool between 4 and 12 hours?

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从变化率到累积的变化

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