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Section 3.1 The integral: accumulation of change
Objectives
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Approximate total change by adding up many small contributions.
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Interpret the definite integral as the signed area between a function and the horizontal axis.
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Connect notation and units to the idea of accumulating tiny pieces.
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Bathtub example? Work example instead? Both?
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NOT doing the bathtub example and avoiding integration of a rate until Chapter 4 could make the FTC more of a surprise!
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Approximation of
\(\DS\sum f(x)\,\Delta x\) by hand using upper and lower sums
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Define \(\DS\int_{a}^{b} f(x)\dd{x}\) as the signed area between \(f(x)\) and the \(x\)-axis between \(x=a\) and \(x=b\text{.}\)
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The
\(\DS\int\) symbol is a long ‘S’ for a smooth Summation.
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The
\(\dd{x}\) is the infinitesimal width of an infinitely thin rectangle.
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