Area Between Curves

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The Area Between Two Curves

To get the area between two curves, we slice the region between them into vertical strips, each of width $\Delta x$ . Denote by $H(x)$ the height at a point $x$ . Since the area of each strip is roughly $H(x)\cdot \Delta x$ , the total area is $\displaystyle\sum_{i=1}^n H(x_i)\, \Delta x$ .

Taking a limit, the area becomes

$\displaystyle\int_a^b H(x)\, dx.$

Notice that sometimes we are given the beginning and ending values of $x$ explicitly, whereas others we have to figure out where two curves meet.