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.
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.
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AP Calculus AB/BC - Unit 6 Overview: Integration and Accumulation of Change
What is the fundamental theorem of calculus?
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