Question 4.7.5: Using the Trapezoidal Rule Compute the Trapezoidal Rule ap...
Using the Trapezoidal Rule
Compute the Trapezoidal Rule approximations with n = 4 (by hand) and n = 8, 16, 32, 64 and 128 (using a program) for \int_0^1 3 x^2 d x.
As we saw in examples 7.1 and 7.2, the exact value of this integral is 1. For the Trapezoidal Rule with n = 4, we have
T_4(f)=\frac{1-0}{(2)(4)}\left[f(0)+2 f\left(\frac{1}{4}\right)+2 f\left(\frac{1}{2}\right)+2 f\left(\frac{3}{4}\right)+f(1)\right]= \frac{1}{8}\left(0+\frac{3}{8}+\frac{12}{8}+\frac{27}{8}+3\right)=\frac{66}{64}=1.03125.
Using a program, you can easily get the values in the accompanying table.
| n | T_n(f) | Error |
| 4 | 1.03125 | 0.03125 |
| 8 | 1.0078125 | 0.0078125 |
| 16 | 1.00195313 | 0.00195313 |
| 32 | 1.00048828 | 0.00048828 |
| 64 | 1.00012207 | 0.00012207 |
| 128 | 1.00003052 | 0.00003052 |
We have included a column showing the error (the absolute value of the difference between the exact value of 1 and the approximate value). Notice that (as with the Midpoint Rule) as the number of steps doubles, the error is reduced by approximately a factor of 4.
Related Answered Questions
Question: 4.5.2
Verified Answer:
Observe that since f(x)=x^{1 / 2}-x^{-2}[/l...
Question: 4.2.7
Verified Answer:
You will need to think carefully about the x’s. Th...
Question: 4.4.6
Verified Answer:
From (4.3), we have
f_{ ave }=\lim _{n \rig...
Question: 4.7.8
Verified Answer:
n
Midpoint Rule
Trapezoidal Rule
Simpson’s Rule
...
Question: 4.4.4
Verified Answer:
From the graph (see Figure 4.19), notice that sin ...
Question: 4.7.2
Verified Answer:
You should confirm the values in the following tab...
Question: 4.3.4
Verified Answer:
The numbers given in the following table are from ...
Question: 4.8.3
Verified Answer:
We can verify this as follows. First, recall that ...
Question: 4.7.3
Verified Answer:
To obtain the desired accuracy, we continue increa...
Question: 4.7.2
Verified Answer:
Approaching the problem graphically, we have five ...