Area Between Curves

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The Fundamental Theorem of Calculus

Three Different Quantities
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Area Between Curves

The Slice and Dice Principle
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The Area Between Two Curves
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Iterated Integrals over Rectangles

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Double Integrals over General Regions

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Summary

If $f(x) \ge g(x)$ on the interval between $x=a$ and $x=b$ , then the area of the region bounded by the curves $y=f(x)$ , $y=g(x)$ , $x=a$ and $x=b$ is $\int_a^b \big(f(x)-g(x)\big)\, dx.$

If $f(x) < g(x)$ , we instead want $\int_a^b \big(g(x)-f(x)\big) \,dx.$ In general, we always want $\int_a^b \hbox{height}(x) \,dx$ , where the height is the larger function value minus the smaller one. This can also be written as $\int_a^b \big|f(x)-g(x)\big| \,dx$ .

If we are not told the beginning and ending values of $x$ , we need to solve $f(x)=g(x)$ to figure them out.

Sometimes it is easier to slice horizontally than vertically. In that case we wind up with an integral $\int_\alpha^\beta \hbox{width}(y)\, dy$ instead of $\int_a^b \hbox{height}(x)\, dx$ , where $\alpha$ and $\beta$ are the smallest and largest values of $y$ .