Area Between Curves

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The Fundamental Theorem of Calculus

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The Indefinite Integral and the Net Change

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Area Between Curves

Computation Using Integration
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The Area Between Two Curves
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Volumes

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Iterated Integrals over Rectangles

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Double Integrals over General Regions

Type I and Type II regions
Examples 1-4
Examples 5-7
Order of Integration

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 do the work as above, but with larger and smaller functions of $y$, and we wind up with an integral $\int_c^d \hbox{width}(y)\, dy$, where $c$ and $d$ are the smallest and largest values of $y$.