To write: The definition for the definite integral of a continuous function from a to b.
Consider the function f which is continuous on interval .
Divide the interval into n subintervals of equal width, .
Let be the end and intermediate points of the subintervals.
Then, the definite interval is the sum of the area of the subintervals. It is expressed as follows:
Here, is any sample point in the subinterval .
To find: The geometric interpretation of if .
The function represents the function in the first quadrant of the graph.
Sketch the curve in the first quadrantas shown in Figure 1.
From Figure 1, the function f is positive when .
The geometric interpretation of refers that the total area which is the region under the graph and above the x-axis on the interval .
To find: The geometric interpretation of , if have both positive and negative values.
The function consists both positive and negative values which means that the graph of forms the region above and below the x-axis.
The graph of function for both positive and negative value of is shown in the below Figure 1.
From the definition from part (a), as takes both positive and negative values the limit of each Riemann sum is also positive, negative or zero.
Thus, the geometric interpretation of refers the signed area under the graph above the x-axis and below the x-axis on the interval .
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