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3-74.

Write and then compute a Riemann sum to estimate the area the curve for $0\le x\le8$ given $f(x) = \sqrt { 64 - x ^ { 2 } }$. Choose the number of rectangles so that your answer will be a good approximation of the area. What is the name of the shape of which you calculated the area? Confirm the accuracy of the Riemann sum by calculating the area geometrically.

General form of a left-endpoint Riemann sum:

$\displaystyle \sum_{i=0}^{n-1}\Delta xf(a+\Delta xi)$

Hint for choosing the number of rectangles, $n$: While you want to choose a large value for $n$, be mindful that you will use that value to compute $\Delta x$.

After all: $\Delta x=\frac{n}{b-a}$

Since $b − a = 8$, it is recommended (though not necessary) that you choose an $n$ that is divisible by $8$.

You should have recognized that this function is a semi-circle with center at the origin and radius $4$. The exact area can be found with geometry.

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Click the link at right for the full version of the eTool: Calc 3-74 HW eTool