My teacher said that the definite integral can be understood as the difference between indefinite integrals. Algebraically the constants cancel each other, but this would mean that indefinite integral represents an area under the graph? How is this possible. If so howd that work?
2026-05-14 12:35:04.1778762104
Linking definite and indefinite integral
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The indefinite integral does represent the area under the graph, but only up to some constant. That is to say that if $F$ is an antiderivative of $f$, $F(x)$ represents the area under $f$ from some starting point to $x$, plus a constant maybe. Thus I can compute the area under the graph between $a$ and $b$ as the difference of the areas from the starting point to $a$ and $b$, and the constants will cancel, giving you simply $F(b) - F(a)$.
Intuituitively, the reason for this is that the derivative (e.g. the rate of growth) of the cumulative area under the curve is simply the height of the curve. More formally, your looking for the fundamental theorem of calculus.