The document discusses the Fundamental Theorem of Calculus. It states that if f(x) is an integrable function and g(x) is an antiderivative of f(x), then the integral of f(x) from a to b is equal to g(b) - g(a). This theorem links the concepts of differentiation and integration by showing that one can find the area under a curve by subtracting the antiderivative evaluated at the bounds. It is an important foundational theorem in calculus.

1. The FUNdamental Theorem of Calculus
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2. Each branch of mathematics has its own fundamental theorem(s).
If you check out the word fundamental in the dictionary,
you will see that it relates to the foundation
or the base
or is elementary.
Fundamental theorems are important foundations
for the rest of the material to follow.
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3. The Fundamental Theorem of Arithmetic
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4. Every integer greater than one
is either prime
or
can be expressed as
an unique product of prime numbers.
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5. Fundamental Theorem of Algebra
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6. Every polynomial equation of degree n
with complex coefficients
has n roots
in the complex numbers.
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7. Fundamental Theorem of Calculus
Here we go! ....................................in the footsteps of Newton and Leibniz
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8. Isaac Newton Gottfried Wilhelm von Leibniz
http://calculustugowar.ytmnd.com/
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9. FUNDAMENTAL THEOREM OF CALCULUS
f(x)
Find the exact area
under the function f(x)
and above the x­axis
in the interval [a, b].
a b
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10. Sketch an antiderivative of f(x) f(x)
a b
g(x) is an antiderivative of f(x)
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12. n subintervals
................
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13. By definition of indefinite integral
g(x1)­g(a) = g'(c1) = f(c1)
Height
c1 c2 c3 cn
Slope Slope of
of = of = Riemann
secant line tangent line rectangle
f(cn)
f(c1) By Mean Value theorem
f(c2)
f(c3)
.............................
n subdivisions
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14. Shazaam!
Math Magic
1. f is an integrable function and
If: 2. g is an antiderivative of f,
b
Then ∫f (x) dx = g(b) ­ g(a)
a
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