Numerical Integration
Here are some numerical approximations to
Z4
1
√xdx =2
3x3/2
4
1
=2
3·8−
2
3·1= 14
3=4.666666 ···.
In the table below, ndenotes the number of subintervals, L(n) is the left-endpoint approximation,
R(n) is the right-endpoint approximation, M(n) is the midpoint approximation, T(n) is the
trapezoid rule approximation, and S(n) is the Simpson’s rule approximation. Each approximation
is accompanied by the difference between the approximation and the true value of the integral.
nL(n) error R(n) error M(n) error T(n) error S(n) error
2 3.871708 −.794959 5.371708 .705041 4.6884769 .0218103 4.6217082 −.0449584 4.6622777 −.0043890
44.280093 −.386574 5.030093 .363426 4.6724008 .0057341 4.6550926 −.0115741 4.6662207 −.0004460
84.476247 −.190420 4.851247 .184580 4.6681230 .0014564 4.6637467 −.0029200 4.6666314 −.0000353
16 4.572185 −.094482 4.759685 .093018 4.6670323 .0003657 4.6659349 −.0007318 4.6666643 −.0000024
32 4.619609 −.047058 4.713359 .046692 4.6667582 .0000915 4.6664836 −.0001831 4.6666665 −1.5·10−7
64 4.643183 −.023484 4.690058 .023391 4.6666896 .0000229 4.6666209 −.0000458 4.6666667 −9.7·10−9
128 4.654936 −.011731 4.678374 .011707 4.6666739 .0000057 4.6666552 −.0000114 4.6666667 −6.1·10−10
256 4.660844 −.005862 4.672523 .005857 4.6666681 .0000014 4.6666638 −.0000029 4.6666667 −3.8·10−11
1. What do you notice about the signs of the errors in the first four methods? Explain your
observations in terms of features of the graph of the function y=√x.
2. What do you notice about the magnitudes of the errors for L(n) and R(n)? How could
L(n) and R(n) be combined to give a more accurate approximation? Where do you find this
approximation in the table?
3. What do you notice about the magnitudes of the errors for M(n) and T(n)? How could
M(n) and T(n) be combined to give a more accurate approximation? Where do you find this
approximation in the table?
4. For each of the methods, what is the ratio of the errors obtained when we double the number of
subintervals? For each method there is a rule of thumb that the error is roughly proportional
to the length of the subinterval raised to a certain power. What powers do you think pertain
to the different methods?