integral calculus adding it all up integral calculus 1
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Integral Calculusadding it all up
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Goal
Compute the signed area under some portion of an arbitrary curve
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Divide and Conquer
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Animatedly
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At Any Given Level
The true area, , is given by
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Summa
This symbol,
is called “summa” or, much more commonly, “the integral sign.”
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More Summa
We sometimes show the limits as part of the summa
which we express as “the integral from a to b of f(t)”.
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Familiar Sights
We’ve seen limits before:
and
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Evaluating an Integral
Remember for derivatives we used the basic relationship
and manipulated that to derive various results.
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Evaluating an Integral (cont.)
For integrals we define the function
and we find its derivative:
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Evaluating an Integral (cont.)
is approximately the area of a little rectangle of height and width . So
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Fundamental Theorem of Calculus
So we have
and
and a doesn’t matter.
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integrand
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Particular Integrals
Suppose F. Working backwards,
So
by which we mean that for any a and b
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Pursuing Similar Reverse Reasoning
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Integration by Parts
Integrate both sides from a to b:
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How Could This Possibly Be Useful?
Consider
So
or
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How To Do Calculus
• Differential calculus• Composition rules• A small number of specific cases• Turn the crank
• Integral calculus• Composition rules• Large books full of specific integrals• Special software• Modest crank-turning 17
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That’s It!
All of Integral and Differential Calculus
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