Trapezoidal Rule MCQ

1. P(0,3), Q(0.5,4) and R(1,5) are three points on the curve defined by f(x). Numerical integration is carried out using both trapezoidal rule and simpson’s rule within limits x = 0 and x = 1 for the curve. The difference between the two results will be

  1. 0
  2. 0.25
  3. 0.5
  4. 1

Answer
Answer. a

2. The error in numerically computing the integral $\int_{0}^{\pi}(\sin x+\cos x)dx$ using the trapezoidal rule with three intervals of equal length between 0 and π is

  1. 0.158
  2. 0.25
  3. 0.20
  4. 0.187

Answer
Answer. d

3. Numerical integration using trapezoidal rule gives the best result for a single variable function, which is

  1. linear
  2. parabolic
  3. logarithmic
  4. hyperbolic

Answer
Answer. a

4. The values of function f(x) at 5 discrete points are given below:

x 0 0.1 0.2 0.3 0.4
f(x) 0 10 40 90 160

Using trapezoidal rule step size of 0.1, the value of $\int_{0}^{0.4}f(x) dx$ is

  1. 44
  2. 22
  3. 11
  4. 33

Answer
Answer. b

5. Using a unit step size, the volume of integral $\int_{1}^{2}(x\ln x)dx$ by trapezoidal rule is

  1. 0.600
  2. 0.493
  3. 0.593
  4. 0.693

Answer
Answer. d

6. The integral $\int_{x_1}^{x_2}(x^2)dx$ with x2 > x1 > 0 is evaluated analytically as well as numerically using a single application of the trapezoidal rule. If I is the exact value of the integral obtained analytically and J is the approximate value obtained using the trapezoidal rule, which of the following statements is correct about their relationship?

  1. J > I
  2. J < I
  3. J = I
  4. insufficient data to determine the relationship

Answer
Answer. a

7. Using the trapezoidal rule, and dividing the interval of integration into three equal subintervals, the definite integral $\int_{1}^{2}\left | x \right |dx$ by trapezoidal rule is

  1. 1.58
  2. 1.25
  3. 1.11
  4. 1.43

Answer
Answer. c

8. The definite integral $\int_{1}^{3}(\frac{1}{x})dx$ is evaluated using trapezoidal rule with a step size of 1. The correct answer is

  1. 1.165
  2. 1.695
  3. 1.213
  4. 1.434

Answer
Answer. a

9. The value of $\int_{2.5}^{4}(\ln x)dx$ calculated using the trapezoidal rule with five subintervals is

  1. 1.2000
  2. 1.4258
  3. 1.7533
  4. 1.6589

Answer
Answer. c

10. The minimum number of equal length subintervals needed to approximate $\int_{1}^{2}(xe^x)dx$ to an accuracy of at least (1/3) x 10-6 using the trapezoidal rule is

  1. 1000e
  2. 1000
  3. 100e
  4. 100

Answer
Answer. a

11. A calculator has accuracy up to 8 digits after decimal place. The value of $\int_{0}^{2\pi}(\sin x)dx$ when evaluated using this calculator by trapezoidal method with 8 equal intervals, to 5 significant digits is

  1. 0.00000
  2. 1.0000
  3. 0.00500
  4. 0.00025

Answer
Answer. a

12. A 2nd degree polynomial, f(x) has values of 1, 4 and 15 at x = 0, 1 and 2, respectively. The integral $\int_{0}^{2}f(x)dx$ is to be estimated by applying the trapezoidal rule to this data. What is the error (defined as “true value-approximate value”) in the estimate?

  1. -4/3
  2. -2/3
  3. 0
  4. 2/3

Answer
Answer. a