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.

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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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