Problem 39E

Explain why or why not

Determine whether the following statements are true and give an explanation or counterexample.

a. The Trapezoid Rule is exact when used to approximate the definite integral of a linear function.

b. If the number of subintervals used in the Midpoint Rule is increased by a factor of 3, the error is expected to decrease by a factor of 8.

c. If the number of subintervals used in the Trapezoid Rule is increased by a factor of 4, the error is expected to decrease by a factor of 16.

Solution:-

Step1

a. The Trapezoid Rule is exact when used to approximate the definite integral of a linear function.

The given statement is true.

The definite integral can be computed by applying linear interpolating formula on each sub interval, and then sum -up them, to get the value of itegral by using the trapezoidal rule.

So, in computing a definite integral of a linear function, the approximated value occurred by using trapezoidal rule is same as the area of the region.

Therefore, by using the trapezoidal rule the value of the definite integral of a linear function is exact.

Step2

b. If the number of subintervals used in the Midpoint Rule is increased by a factor of 3, the error is expected to decrease by a factor of 8.

The given statement is false.

The number of sub-intervals, n increases by a factor of x, then the error decreases by a factor of Midpoint Rule and Trapezoidal Rule.

So, the number of sub - intervals n is increased by a factor of 3, then the error is decreased by a factor of , not 8 for midpoint rule.

Step3

c. If the number of subintervals used in the Trapezoid Rule is increased by a factor of 4, the error is expected to decrease by a factor of 16.

The given statement is true.

The number of sub-intervals, n increases by a factor of x, then the error decreases by a factor of Midpoint Rule and Trapezoidal Rule.

So, the number of sub - intervals n is increased by a factor of 4, then the error is decreased by a factor of , for Trapezoidal Rule.