Problem

Let E be the ellipse given by the equation

Approximate the area of E (to 3 decimal places) using 7 circumscribed rectangles in the first quadrant.

Solution

First, let's examine the graph of E. From the equation, we see that the x axis is the major axis and has length 7. The minor axis is the y axis, with length 4. The ellipse is centered at (0,0).

To approximate the area of E, we'll consider 7 circumscribed rectangles in the first quadrant. Each of these rectangles has width 1. What are their heights?

Given the leftmost x coordinate of an interval, we want to find the corresponding y coordinate, since the circumscribed rectangles touch the ellipse on their upper left corners. We need to find y as a function of x. Since we're limited to the first quadrant, we can do this by solving for y:

In our domain of x values, x is never greater than 7, so the right hand side of this equation is always positive. Thus we can take the square root of both sides and get

Since the square root is always positive, this will give us positive y values when

which is exactly the portion of the ellipse we want to examine (in quadrant 1).

Now that we're equipped with a function to describe the quarter ellipse, we can find the heights of our circumscibed rectangles. We see from the picture that these heights correspond to the leftmost points of each interval along the x axis. Since we have 7 rectangles, these are the points x = 0, 1, 2, 3, 4, 5, 6.

If we evaluate

at each of these points, we get

These are the heights of our rectangles, rounded to 3 decimal places. The width of each rectangle (the distance along the x axis) is 1. so we can find the area of each rectangle and add these areas together.

This is the approximation for the area of the quarter-ellipse. Multiply by 4 to get the area of the entire ellipse, E: