Fundamental Theorem of Calculus: Difference between revisions

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Fundamental Theorem of Calculus (edit)

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Let ''ƒ'' be a continuous real-valued function defined on a closed interval [''a'', ''b'']. Let ''F'' be the function defined, for all ''x'' in [''a'', ''b''], by
Let ''ƒ'' be a continuous real-valued function defined on a closed interval [''a'', ''b'']. Let ''F'' be the function defined, for all ''x'' in [''a'', ''b''], by
:<math>F(x) = \int_a^x f(t)\, dt\,.</math>
:[[File:FTC1.png]]
Then, ''F'' is continuous on [''a'', ''b''], differentiable on the open interval (''a'',&nbsp;''b''), and
Then, ''F'' is continuous on [''a'', ''b''], differentiable on the open interval (''a'',&nbsp;''b''), and


:<math>F'(x) = f(x)\,</math>
:[[File:FTC2.png]]


for all ''x'' in (''a'', ''b'').
for all ''x'' in (''a'', ''b'').
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The fundamental theorem is often employed to compute the definite integral of a function ''&fnof;'' for which an antiderivative ''g'' is known.  Specifically, if ''ƒ'' is a real-valued continuous function on [''a'',&nbsp;''b''], and ''g'' is an antiderivative of ''ƒ'' in [''a'',&nbsp;''b''], then
The fundamental theorem is often employed to compute the definite integral of a function ''&fnof;'' for which an antiderivative ''g'' is known.  Specifically, if ''ƒ'' is a real-valued continuous function on [''a'',&nbsp;''b''], and ''g'' is an antiderivative of ''ƒ'' in [''a'',&nbsp;''b''], then


:<math>\int_a^b f(x)\, dx = g(b)-g(a).</math>
:[[File:FTC3.png]]
   
   
The corollary assumes continuity on the whole interval. This result is strengthened slightly in the following theorem.
The corollary assumes continuity on the whole interval. This result is strengthened slightly in the following theorem.
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Let ''&fnof;'' be a real-valued function defined on a closed interval [''a'', ''b''] that admits an antiderivative ''g'' on [''a'',&nbsp;''b'']. That is, ''ƒ'' and ''g'' are functions such that for all ''x'' in [''a'',&nbsp;''b''],
Let ''&fnof;'' be a real-valued function defined on a closed interval [''a'', ''b''] that admits an antiderivative ''g'' on [''a'',&nbsp;''b'']. That is, ''ƒ'' and ''g'' are functions such that for all ''x'' in [''a'',&nbsp;''b''],


:<math>f(x) = g'(x).\ </math>
:[[File:FTC4.png]]


If ''&fnof;'' is integrable on [''a'',&nbsp;''b''] then
If ''&fnof;'' is integrable on [''a'',&nbsp;''b''] then


:<math>\int_a^b f(x)\,dx\, = g(b) - g(a).</math>
:[[File:FTC5.png]]


Notice that the Second part is somewhat stronger than the Corollary because it does not assume that ''&fnof;'' is continuous.
Notice that the Second part is somewhat stronger than the Corollary because it does not assume that ''&fnof;'' is continuous.
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