Quiz Chapter 07.04: Romberg Method

1. If $I_{n}$ is the value of integral$\, \displaystyle\int_{a}^{b} f(x)dx$ using $n$-segment Trapezoidal rule, a better of the integral can be found using Richardson’s extrapolation as

2. The estimate of an integral of$\, \displaystyle\int_{3}^{19} f(x)dx$ is given as $1860.9$ using $1$-segment Trapezoidal rule. Given $f(7)=20.27, \, f(11)=45.125,$ and $f(14)=82.23$, the value of the integral using $2$-segment Trapezoidal rule would most nearly be

3. The value of an integral$\, \displaystyle\int_{a}^{b} f(x)dx$ given using $1-, \, 2-,$ and $4$-segment Trapezoidal Rule is given as $5.3460, \, 2.7708,$ and $1.7536$, respectively. The best estimate of the integral you can find using Romberg Integration is most nearly

4. Without using the formula for one-segment Trapezoidal Rule for estimating$\, \displaystyle\int_{a}^{b} f(x)dx$ the true error, $E_{t}$, can be found directly as well as exactly by using the formula

• • • $E_{t} = -\dfrac{ \left( b - a \right)^{3}}{12} {f}''(\xi), \, a \leq \xi \leq b$

5. For$\, \displaystyle\int_{a}^{b} f(x)dx$, the true error, $E_{t}$, in $1$-segment Trapezoidal Rule is given by

• • • $E_{t} = - \dfrac{ \left( b - a \right)^{3}}{12} {f}'' ( \xi ), \, a \leq \xi \leq b$

The value of $\xi$ for the integral$\, \displaystyle\int_{2.5}^{7.2} 3e^{0.2x}dx$ is most nearly

6. Given the velocity vs time data for a body

The best estimate for distance covered between $2$s and $10$s by using the Romberg Rule based on the Trapezoidal Rule’s results would be