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How are differentiation and integration connected by the fundamental theorem of calculus?

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transformation of a circular region into an approximately rectangular region — analysis \
This suggests that the same constant (π) appears in the formula for the circumference, 2πr, and in the formula for the area, πr². As the number of pieces increases (from left to right), the “rectangle” converges on a πr by r rectangle with area πr²—the same area as that of the circle. This method of approximating a (complex) region by dividing it into simpler regions dates from antiquity and reappears in the calculus.

The fundamental theorem of calculus establishes a profound inverse relationship between differentiation and integration, serving as the backbone of calculus.12

Articulated independently by Isaac Newton and Gottfried Wilhelm Leibniz, the theorem demonstrates that these two operations are essentially opposites. Specifically, it states that for a continuous function, the derivative of its integral is the original function. In mathematical notation, this is expressed as:3124

$\frac{d}{d t} (\int_{a}^{t} f (u) d u) = f (t)$

Furthermore, the theorem provides the primary method for evaluating definite integrals. It shows that the definite integral of a function over an interval can be calculated by finding its antiderivative—a function whose derivative equals the original function. By linking the calculation of areas under curves (integration) to the determination of rates of change (differentiation), the theorem allows mathematicians and scientists to solve complex problems, such as determining distance from velocity or finding the total accumulation of a quantity from its rate of change.1562