(a)

The function is  and the interval is .

Determine the upper sums and lower sums with .

Consider number of subinterval .

Width of the each rectangle :.

Graph :

Graph the function and upper and lower rectangles.

Observe the graph:

Lower sum refers to the light grey shaded rectangle and upper sum refers to the light grey shaded and dark grey shaded rectangle part.

For the lower sum, consider the minimum values of the function in the interval.

Area of the graph of the function using lower sums is .

For the upper sum, consider the maximum values of the function in the interval.

Area of the graph of the function using upper sums is .

Consider number of subinterval .

Width of the each rectangle :.

Graph :

Graph the function and upper and lower rectangles.

Observe the graph:

Lower sum refers to the light grey shaded rectangle and upper sum refers to the light grey shaded and dark grey shaded rectangle part.

For the lower sum, consider the minimum values of the function in each interval.

Area of the graph of the function using lower sums is .

For the upper sum, consider the maximum values of the function in each interval.

Area of the graph of the function using upper sums is is . 

For , Upper sum is  and lower sum is .

For , Upper sum is  and lower sum is .

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