Problem

Approximate the area of the quarter circle of radius 3 centered at the origin using n circumscribed rectangles. Gives your answer in terms of n.

Solution

We can choose the quarter of the circle that lies in the first quadrant, so we can describe it using the function

Here is what the region looks like, using n = 10 rectangles.

We have n circumscribed rectangles. To approximate the area of the quarter circle using rectangles, we must determine the area of each rectangle and then sum the areas of all rectangles together.

The width of each rectangle is 3/n (since we have n rectangles over an interval of length 3, all of equal width). The height of rectangle 1 is f(0). The height of rectangle 2 is f(3/n), of rectangle 3 is f(6/n), and so on. Continuing to rectangle n, the total area is

Substitute the function definition of f:

This expression can be simplified by factoring the 3/n term:

Using sigma notation, discussed more in 4.2, this can be written as

How close is this to the exact area of the quarter circle? The formula for the area of a circle of radius r is

So the area of a quarter circle of radius 3 equals

or with three decimal places of accuracy,

For each choice of n, the formula for the sum of the areas of the rectangles above gives an approximation to the quarter circle. Note that since the rectangles are circumscribed, the estimate is always larger than the exact area.