Area Sigma Notation, Upper and Lower Sums
Sigma Notation • Definition – a concise notation for sums. • This notation is called sigma notation because it uses the uppercase Greek letter sigma, written as ∑. • The sum of n terms
Using Formulas to Evaluate a Sum • Evaluate the following summation for n = 10, 100, 1000 and 10,000.
Using Formulas to Evaluate a Sum • Now we have to substitute 10, 100, 1000, and 10,000 in for n. • n = 10 the answer is 0.65000 • n = 100 the answer is 0.51500 • n = 1000 the answer is 0.50150 • n = 10,000 the answer is 0. 50015 • What does the answer appear to approach as the n’s get larger and larger (limit as n approaches infinity)?
Area • Finding the area of a polygon is simple because any plane figure with edges can be broken into rectangles and triangles. • Finding the area of a circular object or curve is not so easy. • In order to find the area, we break the figure into rectangles. The more rectangles, the more accurate the area will be.
Approximating the Area of a Plane Region • Use five rectangles to find two approximations of the area of the region lying between the graph of • and the x-axis between the graph of x = 0 and x = 2.
Steps • 1. Draw the graph • 2. Find the width of each rectangle by taking the larger number and subtracting the smaller number. Then divide by the number of rectangles designated. • 3. Now find the height by putting the x values found in number 2 into the equation. • 4. Multiply the length times the height (to find the area of each rectangle). • 5. Add each of these together to find the total area.
Approximating the Area of a Plane Region • Now let’s find the area using the left endpoints. The five left endpoints will involve using the i – 1 rectangle. This answer will be too large because there is lots of area being counted that is not included (look at the graph).
Approximating the Area of a Plane Region • The true area must be somewhere between these two numbers. • The area would be more accurate if we used more rectangles. • Let’s use the program from yesterday to find the area using 10 rectangles, 100 rectangles, and 1000 rectangles. • What do you think the true area is?
Upper and Lower Sums • An inscribed rectangle lies inside the ith region • A circumscribed rectangle lies outside the ith region • An area found using an inscribed rectangle is smaller than the actual area • An area found using a circumscribed rectangle is larger than the actual area • The sum of the areas of the inscribed rectangles is called a lower sum. • The sum of the areas of the circumscribed rectangles is called an upper sum.
Example of Finding Upper and Lower Sums • Find the upper an lower sums for the region bounded by the graph of • Remember to first draw the graph. • Next find the width using the formula
Finding an Upper Sum • Using right endpoints
Limit of the Lower and Upper Sums • Let f be continuous and nonnegative on the interval • [a, b]. The limits as n —›∞ of both upper and lower sums exist and are equal to each other. That is,
Definition of the Area of a Region in the Plane • Let 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