Area of a Plane Region We know how to find the area inside many geometric shapes, like rectangles and triangles. We will now consider finding the area under a curve. The following example will be instructive in developing a general procedure.

Area of a Plane Region Find the approximate area of the region bounded by the graphs of:

The area we are trying to find is enclosed by the four curves.

We begin by dividing the interval from x = 1 to x = 3 into 4 equal subintervals. Each of these subintervals are 0.5 units wide.

In general, if there are n equal subintervals from x = a to x = b, the width of each will be

In general, where i is the i th subinterval, the left endpoint of each interval can be determined by:

Using the formula in the current example, the left end points are:

4 equal subintervals

Draw a rectangle in each subinterval, with the left side of the rectangle touching the curve. Then find the height of each rectangle.

To do this, use the left endpoint of each interval in the function

Find the area of each rectangle. This will be accomplished by multiplying the height (function value) times the width (always 0.5 in this example).

In general, to find the area of the i th rectangle with left endpoints, use the following:

Find the total area of all the rectangles:

Using sigma notation, the sum can be written as:

The sum that we just found is called a Lower Sum since the rectangles are inscribed rectangles (all of them were below the curve).

In general, to find the sum of the areas of all the rectangles using left endpoints, use the following:

Width of Intervals: Summary

Summary Left endpoint of the i th subinterval:

Summary Area of the i th rectangle using left endpoints:

Summary Total area of inscribed rectangles using left endpoints:

Note that the area found using the rectangles is just an approximation of the actual area we wanted.

Since the area found is less than the actual area, let’s repeat the process, only this time using the right endpoints.

The width of each subinterval will be the same as before: Each of the subintervals are 0.5 units wide.

In general, where i is the i th subinterval, the right endpoint of each interval can be determined by:

Using the formula in the current example, the right end points are:

Draw a rectangle in each subinterval, with the right side of the rectangle touching the curve. Then find the height of each rectangle.

To do this, use the right endpoint of each interval in the function

Find the area of each rectangle. This will be accomplished by multiplying the height (function value) times the width (always 0.5 in this example).

In general, to find the area of the i th rectangle with right endpoints, use the following:

Find the total area of all the rectangles:

Using sigma notation, the sum can be written as:

The sum that we just found is called a Upper Sum since the rectangles are circumscribed rectangles (all of the tops of the rectangles are above the curve).

In general, to find the sum of the areas of all the rectangles using right endpoints, use the following:

Width of Intervals: Summary

Summary Right endpoint of the i th subinterval:

Summary Area of the i th rectangle using right endpoints:

Summary Total area of circumscribed rectangles using right endpoints:

Rather than calculating the area of each rectangle and finding the sum, we can use the formulas.

Note that this is the same value found earlier in calculating the sum of the areas of the circumscribed rectangles.

Once again the area found using the rectangles is just an approximation of the actual area we wanted.

In this case the approximation turns out to be larger than the actual area.

Area Left EndpointsRight Endpoints

Conclusion We know the actual area is between 6.75 sq units and sq units. This isn’t very “close”. How do we get a better estimate? There are two possibilities: 1.Use rectangles that are closer to estimating the area. In the current example, using the midpoint of the interval would give a better estimate. 2.Use more rectangles. It can be shown that as the number of rectangles approaches infinity, the area will be exact.

Definition of the Area of a Region in the Plane Let function f be continuous and nonnegative on the interval [ a, b ] The area of the region bounded by the graph of f, the x -axis, and the vertical lines x = a and x = b is given by:

Definition of the Area of a Region in the Plane This is stating that c i can be any point in the interval, including the left or right endpoints. This is the width of each interval.

Definition of the Area of a Region in the Plane Area of Rectangle (height times width) Add up all the areas of all the rectangles Let the number of rectangles approach infinity