To show:The function is not integrable on .
Given information:The function is if and .
It is known that a function for the interval should be divided into subintervals of equal width:
Consider that the number of rectanglesof subintervals for be . Substitute 0 for and 1 for in the above formula to find the width of the rectangle.
If left endpoints is used then the height of the first rectangle be 0 because .Now, the second rectangle made arbitrarily large value. Consider a sample point that is closer to 0 for the first rectangle, such as .
The area of the first rectangle is as follows:
The function is positive on the interval. If is a finite number, then the whole sum is greater than the area of the first rectangle. So,
Now, increase the value of toward infinity to get:
The first term and the entire sum tendtowards infinity. A function is not integrable on a certain interval if the sum tend towards .
Hence, it is proved that the function is not integrableon .
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