5.6. NUMERICAL INTEGRATION
323
Activity 5.15.
In this activity, we explore the relationships among the errors generated by left, right,
midpoint, and trapezoid approximations to the definite integral
2
1
1
x 2 dx
(a) Use the First FTC to evaluate
2
1
1
x 2 dx exactly.
(b) Use appropriate computing technology to compute the following approximations for
2
1
1
x 2 dx: T 4 , M 4 , T 8 , and M 8 .
(c) Let the error of an approximation be the difference between the exact value
of the definite integral and the resulting approximation. For instance, if we
let E T,4 represent the error that results from using the trapezoid rule with 4
subintervals to estimate the integral, we have
E T,4 =
2
1
1
x 2 dx − T 4 .
Similarly, we compute the error of the midpoint rule approximation with 8
subintervals by the formula
E M,8 =
2
1
1
x 2 dx − M 8 .
Based on your work in (a) and (b) above, compute E T,4 , E T,8 , E M,4 , E M,8 .
(d) Which rule consistently over-estimates the exact value of the definite integral?
Which rule consistently under-estimates the definite integral?
(e) What behavior(s) of the function f (x) =
1
x 2 lead to your observations in (d)?
⊳
Comparing the Midpoint and Trapezoid Rules
We know from the definition of the definite integral of a continuous function f , that if
we let n be large enough, we can make the value of any of the approximations L n , R n ,
and M n as close as we’ d like (in theory) to the exact value of
b
a
f (x) dx. Thus, it may
be natural to wonder why we ever use any rule other than L n or R n (with a sufficiently
large n value) to estimate a definite integral. One of the primary reasons is that as n → ∞,
△x =
b−a
n → 0, and thus in a Riemann sum calculation with a large n value, we end up
multiplying by a number that is very close to zero. Doing so often generates roundoff error,
as representing numbers close to zero accurately is a persistent challenge for computers.
Hence, we are exploring ways by which we can estimate definite integrals to high levels
of precision, but without having to use extremely large values of n. Paying close attention
323
Activity 5.15.
In this activity, we explore the relationships among the errors generated by left, right,
midpoint, and trapezoid approximations to the definite integral
2
1
1
x 2 dx
(a) Use the First FTC to evaluate
2
1
1
x 2 dx exactly.
(b) Use appropriate computing technology to compute the following approximations for
2
1
1
x 2 dx: T 4 , M 4 , T 8 , and M 8 .
(c) Let the error of an approximation be the difference between the exact value
of the definite integral and the resulting approximation. For instance, if we
let E T,4 represent the error that results from using the trapezoid rule with 4
subintervals to estimate the integral, we have
E T,4 =
2
1
1
x 2 dx − T 4 .
Similarly, we compute the error of the midpoint rule approximation with 8
subintervals by the formula
E M,8 =
2
1
1
x 2 dx − M 8 .
Based on your work in (a) and (b) above, compute E T,4 , E T,8 , E M,4 , E M,8 .
(d) Which rule consistently over-estimates the exact value of the definite integral?
Which rule consistently under-estimates the definite integral?
(e) What behavior(s) of the function f (x) =
1
x 2 lead to your observations in (d)?
⊳
Comparing the Midpoint and Trapezoid Rules
We know from the definition of the definite integral of a continuous function f , that if
we let n be large enough, we can make the value of any of the approximations L n , R n ,
and M n as close as we’ d like (in theory) to the exact value of
b
a
f (x) dx. Thus, it may
be natural to wonder why we ever use any rule other than L n or R n (with a sufficiently
large n value) to estimate a definite integral. One of the primary reasons is that as n → ∞,
△x =
b−a
n → 0, and thus in a Riemann sum calculation with a large n value, we end up
multiplying by a number that is very close to zero. Doing so often generates roundoff error,
as representing numbers close to zero accurately is a persistent challenge for computers.
Hence, we are exploring ways by which we can estimate definite integrals to high levels
of precision, but without having to use extremely large values of n. Paying close attention