Fundamental Theorem of Calculus

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In this lesson, we introduce the Fundamental Theorem of Calculus.

What is the Fundamental Theorem of Calculus?

The Fundamental Theorem of Calculus, Part 1

If f is continuous on [a, b], then the function g is defined by

is continuous on [a, b] and differentiable on (a, b), and g'(x) = f(x)

The Fundamental Theorem of Calculus, Part 2

If f is continuous on [a, b], then

where F is any antiderivative of f, that is, a function such that F ’ = f

Example:

Find the area under the parabola y = x² from 0 to 1.

Solution:

An antiderivative of f (x) = x² is

Use Part 2 of the Fundamental Theorem to find the required area A.

The Fundamental Theorem of Calculus
The fundamental theorem of Calculus is an important theorem relating antiderivatives and definite integrals in Calculus. The fundamental theorem of Calculus states that if a function f has an antiderivative F, then the definite integral of f from a to b is equal to F(b)-F(a). This theorem is useful for finding the net change, area, or average value of a function over a region.

The Fundamental Theorem of Calculus Part 1 The Fundamental Theorem of Calculus Part 2

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