(a)

The definite integral is .

Determine by left end points:

Observe the graph of .

Number of subintervals are .

Width of the interval is  :.

Left end points are and .

Riemann sum is .

.

Substitute corresponding function values in above expression from the graph.

Area under the graph of the function using left endpoints is .

.

Determine using right endpoints.

Number of subintervals are .

Width of the interval is  :.

Right end points are and .

Riemann sum is .

.

Substitute corresponding function values in above expression observe from the graph.

Area under the graph of the function using right endpoints is .

.

Determine by Mid points:

Number of subintervals are .

Width of the interval is  :.

, where .

Mid points are and .

Riemann sum is .

.

Substitute corresponding function values in above expression from the given graph.

Area under the graph of the function using mid points is .

.

(b)

Consder the graph is approximately equal to sum of the area of the right angle triangle  and rectangle.

Area of the triangle is .

From the graph, and .

Therefore, area of the graph is

Area of the rectangle is .

.

Total area under the graph is

Actual area of the graph is 8 units.

Now compare and values with actual area value.

Therefore, is an underestimate and are over estimates.

(c)

Find :

Trapezoidal rule:

Where, .

Substitute corresponding values in the above expression.

(d)

For any value of , the values of and .

Observe the results in step 1, 2, and 3.

. 

(a) 

, and

(b)

  is an underestimate and are over estimates.  

(c)  

(d)

.