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3-74.

Write and then compute a Riemann sum to determine the area $A(f,-4 ≤ x ≤ 4)$ where $f(x)$ is the function below. 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.

$f ( x ) = \sqrt { 16 - x ^ { 2 } }$

General form of a left-endpoint Riemann sum:

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 $Δx$.
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.

Use the eTool below to visualize this problem.
Click the link at right for the full version of the eTool: Calc 3-74 HW eTool