Table of values of an increasing function  is given.

Determine the integral .

Number of subintervals are .

(a)

Determine the integral using right end approximation.

Here and .

Width of the interval is  :.

Right end points are .

Substitute corresponding function values in above expression from the table.

Integral value using right end approximation is .

(b)

Determine the integral using left end approximation.

Width of the interval is:.

Number of subintervals are .

Left end points are .

.

.

Substitute corresponding function values in above expression from the table.

Integral value using left end approximation is .

(c)

Determine the integral using mid point approximation.

Width of the interval is:.

Number of subintervals are .

Mid points are .

, where .

Substitute corresponding function values in above expression from the table.

Integral value using mid point approximation is .

If the function is increasing function, then the left end approximation indicates underestimate of actual integral value.

Right end approximation indicates over estimate of actual integral value.

Therefore, the left end approximation gives less than exact integral value and right end approximation gives greater than exact integral value.

(a)

Integral value using right end approximation is .

(b)

Integral value using left end approximation is .

(c)

Integral value using mid point approximation is .

The left end approximation gives less than exact integral value and right end approximation gives greater than exact integral value.