Examination Paper I

The goal of this paper is to assess your understanding of the definite integral. You have two choices in terms of problems. The second set are more substantial and involve heavy use of the computer.

  • Six Problems from Section I
  • Two Problems from Section II

Guidance on which problems to complete and how to complete them will depend on your present interest and comfort with the definite integral.

Section I: Approximation Methods

Rectangles

Problem 1. Using the image above, with four subintervals as shown, and an arbitrary function \(f(x)\), determine and explain the formula requested. Your formula should be in terms of the left endpoint of the subinterval \(a\), the right endpoint of the interval \(b\), and the number of subdivisions of the interval \(n\):

  1. Rectangles built on the left endpoint of each subinterval
  2. Rectangles built on the right endpoint of each subinterval
  3. Rectangles built on the midpoint of each subinterval

Trapezoids

Problem 2. Determine the formula using the image above, with four subintervals, an arbitrary function \(g(x)\), and trapezoids connected by the top of each vertical bar, for example, the first trapezoid would be determined by \(x_1, x_2, f(x_1), \text{and}~ f(x_2)\).

Parabolas

Problem 3. Derive and use Simpson’s rule to approximate one of the integrals in section below. This is not for the weak at heart.

Using the Formulas

Problem 4. Use the left, right, or midpoint formulas to approximate the area under the given curves on the given intervals, with the appropriate n values.

  1. \(f(x) = x^2 + x\) on \([0, 4]\) with 8 rectangles.
  2. \(g(x) = 2 + \sin{x}\) on \([0, 3\pi]\) with 6 rectangles.
  3. Solve problem 1 using trapezoids instead of rectangles.
  4. Solve problem 2 using trapezoids instead of rectangles.

Exact Formula

Here, we focus on situations where we allow the number of rectangles to approach infinity. We allow the use of rules of integration and computer evaluation of integrals. The following questions deal with typical topics from large scale calculus examinations.

\[\displaystyle \int_a^b f(x) ~ dx\]

Area Under Curve

Problem 5. Use the appropriate rules for integration or the computer to evaluate the given definite integral. Interpret the problem in terms of area under the curve.

  1. \(f(x) = x^4 - 3x^2 + 5\) on \([1, 5]\)
  2. \(f(x) = \sin{x} - e^x\) on \([\pi, 4\pi]\)
  3. \(f(x) = x^3\cos{x}\) on \([0, 2\pi]\) (Hint: Advanced Formula Required by Hand)

Area Between Curves

Problem 6. Solve these problems dealing with the area between curves. Be sure you show how all answers were determined.

  1. \(f(x) = \sin{x}\) and \(g(x) = \cos{x}\)
  2. \(f(x) = 2-x^2\) and \(g(x) = x\)
  3. \(f(x) = -x^2 + 3x\) and \(g(x) = 2x^3 - x^2 - 5x\)

Volume of Solids formed by Revolution

Problem 7. Find the volume of the solid formed by rotating each of the regions above in the Area Between Curves about the \(x\)-axis.

Economics Applications

Problem 8. Answer the problems below dealing with the Gini Index as a measure for inequality.

  1. The table below shows how total energy produced and total energy consumed was divided among the 25 largest producers and users of energy in 1963. Draw the corresponding Lorenz curve and perfect equality line. Was production of energy more or less equally divided than was use of energy? Why? Interpret the Gini Index in this context.
Fraction of Countries Energy Used Energy Produced
0.0 0.00 0.00
0.2 0.027 0.032
0.4 0.065 0.084
0.6 0.135 170
0.8 0.266 0.3031
1.0 1.000 1.0000
  1. The table below shows how gross national product (GNP) is distributed among the countries of the world:
Fraction of Countries Fraction of GNP
0.0 0.0
0.1 0.001
0.2 0.002
0.3 0.005
0.4 0.010
0.5 0.018
0.6 0.028
0.7 0.058
0.8 0.11
0.9 0.21
1.0 1.00
  1. Determine the Lorenz Curve for this data using the model you deem most appropriate.
  2. Draw the Lorenz Curve and the curve for completely unequal distribution on the same axes.
  3. Is the GNP divided equally or unequally among the countries of the world?

Problem 9. The concept of Gini Coefficients to measure inequality is a popular topic in Classical Economics type textbooks. For example, Paul Samuelson’s Economics. Consult an economics textbook for additional examples, and rewrite some examples to incorporate finding the equation of the curve and using integration to determine the Gini value.

Problem 10. Answer the following questions pertaining to the Consumer and Producer surplus.

  1. Find the equilibrium price and quantity and consumer surplus for the given supply and demand functions.
  2. \(s(q) = \frac{q}{27,000} \quad D(q) = 10 - \frac{q}{3,000}\)
  3. \(s(q) =3q \quad D(q) = 6 - 3q \quad \text{where q is in 10,000's of units}\)
  4. Discrimination by consumers means buying each additional unit \((\Delta q_i)\) at cost (which is \(S(q_i)\). Total revenue now become equal to this since
\[\sum p_i \Delta q_i = \sum S(q_i) \Delta q_i \approx \int_0^{q\star} s(q)dq\]

Make an argument that this represents the cost to the producers of the goods sold on the market. Thus arguing that the area below the equilibruim price line and above the supply curve, \(p = S(q)\) on producers in a competitive market, the producer’s surplus. Make a visualization of this concept. Give the integral formula for the producer’s surplus. Calculate the producer’s surplus for each of the problems 1a and 1b.

Section II: Computing and Further Applications

Problem 11: Archimedes use of exhaustion to approximate pi can be framed as a recurrence relationship. Your task is to explain this relationship using the discussion in Edwards Historical Origins of the Calculus, and to write a computer program that carries out approximations to pi to a given number of sides of polygons.

Problem 12: Archimedes determination of the Area under a Parabola can be framed as sequence of sums of triangular areas. Write a short program that allows users to input the number of divisions under the Parabola, and outputs the approximation with a visualization of the region.

Problem 13: Write a computer program that takes as input a function, left endpoint, right endpoint, and number of subintervals and returns the approximation for rectangles and trapezoids as well as the exact area. You should have a visualization with three subplots that demonstrate the approximations.

Probability and Statistics

We need better materials addressing probability and statistics and their connections to introductory integration techniques. These problems will deal with developing short introductory videos and/or other media tutorials for important ideas from Probability and Statistics.

Problem 14: What is a probability distribution? What is the difference between discrete and continuous probability distributions? Explain these in terms of a specific example? You should frame these in terms of summations and definite integrals.

Problem 15: What is a Poisson Model and what kind of situations do we use it for?

Problem 16: What is an exponential random variable? What are some examples of situations where we would use this?

Problem 17: What is the normal distribution? What are some examples of situations where we would us this? What is special about the standard deviation and areas under the curve?

Problem 18: What is the difference between a probability distribution and a cumulative probability distribution? Please use an example to demonstrate.

Problem 19: Textbook Problems. Read Gilbert Strang’s Section on Probability and Calculus. Answer questions #24 - 36.

Figure Source Code

In [64]:

%matplotlib inline
import matplotlib.pyplot as plt
import numpy as np

fig = plt.figure('Figure 1', figsize = (10, 7))
x = np.linspace(-2, 2, 1000)
def f(x):
    return 2*np.sin(x**3) + 4
plt.plot(x, f(x), color = 'black')
plt.axhline(color = 'black')
plt.fill_between(x, f(x), alpha = 0.2, color = 'green')
width = 0.5
a = [-2 + i*width for i in range(1, 6)]
plt.vlines(a, 0, [f(i) for i in a])
plt.xticks([-1.5, -1.0, -0.5, 0.0, 0.5], ['$a = x_1$', '$x_2$', '$x_3$', '$x_4$', '$x_5 = b$'], size = 18)
plt.annotate('Area Under the Curve $f(x)$', xy=(0, 2), xytext=(-2.0, 5), arrowprops=dict(facecolor='black'), size = 18)
plt.title('Approximating the Area Under the Curve\nusing Rectangles, Trapezoids, and Parabolas', loc = 'left', size = 12)
plt.savefig('images/exam_approx.png')
_images/11-Exam-I_7_0.png