Evaluating a Definite Integral as a Limit In Exercises 5-10, evaluate the definite integral by the limit definition.
To calculate: Definite integral of by the limit definition.
Formula for the definite integral of from to : where, is the lower limit of integration and is the upper limit of integration.
Formula for the sum of a constant times: where is a constant.
Formula for the sum of first terms:
Formula for the sum of squares of first terms:
Calculation: Function can be integrated on the interval because it is continuous on .
Definition of integrability implies that any partition whosenorm approaches can be used to determine the limit.
For computational convenience, define by subdividing into subintervals of equal width as below:
Choosing as the right endpoints of each subintervals produces,
So, the definite integral is:
Use the values of and .
So, the expression becomes,
Factor out from the sum
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