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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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