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Fundamental Theorem of Calculus

Definition

The Fundamental Theorem of Calculus states that if f(x) is continuous on an interval [a, b] and F(x) is any antiderivative of f(x), then ∫[a,b] f(x) dx = F(b) - F(a). In simpler terms, it relates differentiation and integration by showing that finding the area under a curve can be done by evaluating its antiderivative at two points.

Analogy

Imagine you have two points on opposite ends of a river. You want to know how much water has flowed through that section. Instead of measuring every single drop, you can simply subtract the initial water level from the final water level. Similarly, instead of adding up infinitely many tiny rectangles under a curve, we can evaluate its antiderivative at two points to find its area.

Related terms

Antiderivative: An antiderivative (also known as an indefinite integral) reverses differentiation and gives us back the original function before it was differentiated.

Definite Integral: A definite integral is used to find the exact area under a curve between two given points. It is represented by ∫[a,b] f(x) dx, where a and b are the limits of integration.

Riemann Sum: A Riemann sum is an approximation of the area under a curve using rectangles. It involves dividing the interval into subintervals and evaluating the function at specific points within each subinterval.