Estimation and Limits -- Area -- Reference

Consider the region shown above. This region is bounded by the four sides below:

• The x-axis.
• The vertical line x = a.
• The vertical line x = b.
• The curved top edge y = f(cx).

We can approximate the area of this region by dividing the interval [a, b] into n strips of equal width

as shown in the figure below

We mark the following points along the x-axis.

Next we estimate the area of each strip by a rectangle. The i-th strip goes from x(i - 1) to x(i) and we use a rectangle whose height, f(x(i - 1)) is determined by the left edge. The area of this rectangle is its height multiplied by its width -- f(x(i - 1)) h. We estimate the area of the entire region by summing the estimates for the n strips.

Notice that we might just as well have estimated the area of each strip by a rectangle whose height was determined by the right edge of the strip -- f(x(i)) .

In any case by using a very large number of strips we can get a very good estimate for the area of our region. The exact area is the limit of these estimates.

Check Your Understanding

For each of the following problems find the best estimate that you can for the area of the region bounded by the four sides. Use both estimates based on rectangles whose height is determined by the left edge of each strip and estimates based on rectangles whose height is determined by the right edge of each strip.

• The x-axis.
• The vertical line x = a.
• The vertical line x = b.
• The curved top edge y = f(x).

in a reasonable amount of time.

1. f(x) = sin(x), a = 0, b = pi/2.

   answer

2. f(x) = sin(x), a = 0, b = pi.

   answer

3. f(x) = x^2 + 1, a = 0, b = 1.

   answer

4. f(x) = x^2 + 1, a = 0, b = 4.

   answer

5. f(x) = cos(x), a = 0, b = pi/2.

   answer

6. f(x) = 2^x, a = 0, b = 2.

   answer