
Concept explainers
(a) Draw two typical curves y = f(x) and y = g(x), where f(x) ≥ g(x) for a ≤ x ≤ b. Show how to approximate the area between these curves by a Riemann sum and sketch the corresponding approximating rectangles. Then write an expression for the exact area.
(b) Explain how the situation changes if the curves have equations x = f(y) and x = g(y), where f(y) ≥ g(y) for c ≤ y ≤ d.
(a)

To Draw: the two typical curves
To define: A Riemann sum that approximates the area between the two typical curves with drawing of the corresponding approximating rectangles and exact area between the two typical curves and the expression for the exact area.
Explanation of Solution
Consider the two curves
Here, the top curve function is
Assume f and g are continuous function and
Here, the lower limit is a and the upper limit is b.
Show the approximate ith strip rectangle with base
Sketch the two typical curves
Refer to figure 1.
The two typical curves
The expression for the exact area is
Divide the area between the two typical curves into n strips of equal width and take the entire sample points to be right endpoints, in which
Sketch thecorresponding approximating rectangles as shown in Figure 2.
The better and better approximation occurs in
Thus, the Riemann sum with the sketch of corresponding approximating rectangles and the exact area between the two typical curves shown.
Therefore, the approximation of the area between the two typical curves using Riemann sum with the sketch of the corresponding approximating rectangles and the sum of the areas corresponding approximating rectangles is the exact area.
(b)

To Draw: The two typical curves with the changing the situation as
To define: The situation if the curves changes from
The expression for the exact area is
Explanation of Solution
Consider the two curves
Here, the right curve function is
Assume f and g are continuous function and
Here, the bottom limit is c and the top limit is d.
Sketch the two typical curves
Thus, the two typical curves
Normally the height calculated from the top function minus bottom one and integrating from left to right. Instead of normal calculation, use “right minus left” and integrating from bottom to top. Therefore the exact area, A written as
Therefore, the changes of the situation if the curves have equations
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