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26B First Fundamental Theorem

The First Fundamental Theorem

of Calculus

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26B First Fundamental Theorem

The First Fundamental Theorem of Calculus

Let f  be continuous on [a,b] and let x  be a value in (a,b). Then

Theorem Comparison Property

If f and g are integrable on [a,b] and if f(x)!g(x) for all x  on [a,b],

then

Theorem Boundless Property

If f is integrable on [a,b] and m!f(x)!M  for all a on [a,b],

then m(b-a)!  !M(b-a)

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26B First Fundamental Theorem

Theorem Linearity of the Definite Integral

If f  and g  are integrable on [a,b] and k  is a real number, then

(i)

and(ii)

EX 1 Suppose

Calculate .

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26B First Fundamental Theorem

EX 3 Find

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26B First Fundamental Theorem