Approximation of definite integrals using the trapezoidal rule：練習問題

Estimate the value of the definite integral \(\int_{1}^{3} \frac{1}{x} dx\) using the trapezoidal rule with 2 sub-intervals of equal width. Give your answer correct to 3 decimal places.

The table below shows some values of a continuous function \(f(x)\):

\(x\): 0.0, 0.5, 1.0, 1.5, 2.0
\(f(x)\): 2.0, 2.4, 3.1, 4.2, 6.0

Using the trapezoidal rule with 4 strips, estimate the value of \(\int_{0}^{2} f(x) dx\).

Consider the definite integral \(\int_{0}^{2} \frac{1}{x+1} \, dx\).

Estimate the value of the integral using the trapezoidal rule with 4 strips. Determine whether this estimate is an over-estimate or an under-estimate by analyzing the concavity of the integrand over the given interval.

Consider the definite integral \(I = \int_{0}^{1} \ln(x+1) dx\). If the trapezoidal rule is used to estimate the value of \(I\), which of the following statements is true?

Let \(I = \int_{0}^{1} e^{-x} dx\). Let \(T_2\) and \(T_4\) be the estimates of \(I\) obtained by the trapezoidal rule with 2 and 4 sub-intervals of equal width respectively. Which of the following relationships is correct?

Estimate the value of \(\int_{0}^{2} (x^3+1) \, dx\) using the trapezoidal rule with 2 strips.

Estimate the value of \(\int_{1}^{2} \frac{4}{x^2} \, dx\) using the trapezoidal rule with 4 sub-intervals. Give your answer correct to 3 decimal places.

Consider the definite integral $$\int_{0}^{2} \sqrt{1+x^2} \, dx$$. Using the trapezoidal rule with 4 strips, estimate its value. Justify whether this estimate is an over-estimate or an under-estimate by analyzing the concavity of the integrand over the given interval.

Consider the definite integral \(I = \int_1^3 \sqrt{3x-2} \, dx\).
(a) Find the width of each sub-interval if the interval \([1, 3]\) is divided into 4 equal strips.
(b) Use the trapezoidal rule with 4 strips to estimate the value of \(I\). Give your answer correct to 3 decimal places.

The velocity of a particle, \(v(t)\) in m/s, is recorded at 1-second intervals as shown in the table below:

\(\begin{array}{|c|c|c|c|c|c|} \hline t \text{ (s)} & 0 & 1 & 2 & 3 & 4 \\ \hline v(t) \text{ (m/s)} & 0.00 & 2.08 & 3.30 & 4.16 & 4.83 \\ \hline \end{array}\)

(a) Estimate the total distance traveled by the particle from \(t=0\) to \(t=4\) using the trapezoidal rule with 4 strips.
(b) A scientist models the velocity using the function \(f(t) = 3\ln(t+1)\). Find \(f''(t)\).
(c) Based on the model \(f(t)\), determine whether the estimate in (a) is likely to be an over-estimate or an under-estimate of the true distance traveled.