WisIIA-HC01.pdf

# The definite integral: definition and interpretation This section details the definition of the definite integral, its graphical meaning as area, and its fundamental properties. Add some stuff ### 1.1 Definition of the definite integral The definite integral of a continuous function $f$ on an interval $\[a, b\]$ is defined as: $$ \\int\_{a}^{b} f(x) , dx = \[F(x)\]\_{a}^{b} = F(b) - F(a) $$ where $F$ is an antiderivative of $f$ on $\[a,b\]$. When finding $F(x)$, the constant of integration can be omitted [3](#page=3) [4](#page=4). **Example:**$$ \\int\_{0}^{1} (2x + 1) , dx = \[x^2 + x\]\_{0}^{1} = (1^2 + 1) - (0^2 + 0) = 2 $$ ### 1.2 Graphical interpretation as area When the function $f$ is positive on the interval $\[a, b\]$, the definite integral represents the area $S$ under the curve of $f(x)$ from $a$ to $b$: $$ S = \\int\_{a}^{b} f(x) , dx $$ **Exercise:** Calculate the area of the region bounded by the graph of $y = 2x + 1$, the x-axis, the y-axis, and the line $x = 1$ [6](#page=6). However, if the function $f$ is negative on $\[a, b\]$, the definite integral will yield a negative value. In such cases, the area $S$ is the negative of the definite integral: $$ S = -\\int\_{a}^{b} f(x) , dx $$ It is crucial to understand that the definite integral does not always represent the geometric area when the function takes on both positive and negative values. For instance, $\\int\_{-2}^{2} x^3 , dx$ is not equal to the geometric area between the curve and the x-axis, which is 8. The reason is that the integral accounts for the signed area, where areas above the x-axis are positive and areas below are negative [7](#page=7). ### 1.3 Properties of the definite integral The definite integral possesses several key properties: 1. **Reversing the limits of integration:**$$ \\int\_{a}^{b} f(x) , dx = -\\int\_{b}^{a} f(x) , dx $$ 2. **Integral over a point:**$$ \\int\_{a}^{a} f(x) , dx = 0 $$ 3. **Additivity of integration intervals:** For any $c$ between $a$ and $b$: $$ \\int\_{a}^{b} f(x) , dx = \\int\_{a}^{c} f(x) , dx + \\int\_{c}^{b} f(x) , dx $$ 4. **Constant multiple rule:**$$ \\int\_{a}^{b} \\alpha f(x) , dx = \\alpha \\int\_{a}^{b} f(x) , dx \\quad (\\alpha \\in \\mathbb{R}) $$ 5. **Sum/Difference rule:**$$ \\int\_{a}^{b} (f(x) + g(x)) , dx = \\int\_{a}^{b} f(x) , dx + \\int\_{a}^{b} g(x) , dx $$ **Exercise:** Calculate: $$ \\int\_{0}^{\\ln 1} e^{-x^2} , dx $$ ### 1.4 Area between two curves The area between two curves, $f(x)$ and $g(x)$, can be calculated by integrating the difference between the upper and lower curves over the relevant intervals. If the curves intersect at one point $c$ within the interval $\[a, b\]$, the area is calculated as: $$ \\text{OPP} = \\int\_{a}^{c} (g(x) - f(x)) , dx + \\int\_{c}^{b} (f(x) - g(x)) , dx $$ assuming $g(x) \\geq f(x)$ on $\[a, c\]$ and $f(x) \\geq g(x)$ on $\[c, b\]$ [11](#page=11). If there are multiple intersection points (e.g., $a < c < d < b$), the area calculation involves summing the areas between the curves in each sub-interval, ensuring the integrand is always the difference between the upper and lower curve: $$ \\text{OPP} = \\int\_{a}^{c} (f(x) - g(x)) , dx + \\int\_{c}^{d} (g(x) - f(x)) , dx + \\int\_{d}^{b} (f(x) - g(x)) , dx $$ **Example:** Calculate the area between two curves: $$ \\text{Opp.} = \\int\_{0}^{1} (x - x^3) , dx + \\int\_{1}^{2} (x^3 - x) , dx = \\frac{1}{4} + \\frac{9}{4} = \\frac{5}{2} $$ ### 1.5 Approximation of the definite integral The definite integral can be approximated by summing the areas of a series of rectangles. If $\\Delta x$ is the width of each rectangle (where $\\Delta x = \\frac{b-a}{n}$) and $f(x\_i)$ is the height of the $i$\-th rectangle (with $x\_i$ being a value within the $i$\-th sub-interval), then the area $S$ can be expressed as the limit of the sum of these rectangular areas as the number of rectangles approaches infinity: $$ S = \\lim\_{n \\to \\infty} \\sum\_{i=1}^{n} f(x\_i) \\Delta x = \\int\_{a}^{b} f(x) , dx $$ This process illustrates the fundamental connection between summation and integration. * * * # Economic applications of the definite integral The definite integral finds practical applications in economics by quantifying areas under curves, particularly for concepts like consumer and producer surplus, and the Gini coefficient [15](#page=15) [20](#page=20). ### 2.1 Consumer and producer surplus Consumer surplus (CS) and producer surplus (PS) are economic measures that utilize definite integrals to represent the benefits consumers and producers receive from market transactions [15](#page=15). #### 2.1.1 Consumer surplus (CS) Consumer surplus represents the difference between what consumers are willing to pay for a good or service and what they actually pay. It is calculated as the area between the demand curve ($D(q)$) and the market price ($p^{\\ast}$) up to the equilibrium quantity ($q^{\\ast}$) [15](#page=15). The formula for consumer surplus is: $$CS = \\int\_{0}^{q^{\\ast}} (D(q) - p^{\\ast}) , dq$$ This can be further expressed as: $$CS = \\int\_{0}^{q^{\\ast}} D(q) , dq - p^{\\ast}q^{\\ast}$$ > **Tip:** Visually, consumer surplus is the area below the demand curve and above the horizontal line representing the market price, from the origin to the equilibrium quantity. #### 2.1.2 Producer surplus (PS) Producer surplus represents the difference between the price producers receive for a good or service and the minimum price they are willing to accept (their cost of production). It is calculated as the area between the market price ($p^{\\ast}$) and the supply curve ($S(q)$) up to the equilibrium quantity ($q^{\\ast}$) [15](#page=15). The formula for producer surplus is: $$PS = \\int\_{0}^{q^{\\ast}} (p^{\\ast} - S(q)) , dq$$ This can be further expressed as: $$PS = p^{\\ast}q^{\\ast} - \\int\_{0}^{q^{\\ast}} S(q) , dq$$ > **Tip:** Visually, producer surplus is the area above the supply curve and below the horizontal line representing the market price, from the origin to the equilibrium quantity. #### 2.1.3 Example: Calculating CS and PS To illustrate, consider a scenario with given demand and supply functions where the equilibrium quantity ($q^{\\ast}$) and market price ($p^{\\ast}$) are known. The calculation involves setting up and evaluating the definite integrals for CS and PS. For instance, if the demand function is $D(q) = 60 - q$ and the supply function is $S(q) = q$, and the equilibrium occurs at $q^{\\ast} = 20$ and $p^{\\ast} = 20$, then [16](#page=16) [17](#page=17) [18](#page=18) [19](#page=19): Consumer Surplus: $CS = \\int\_{0}^{20} ((60 - q) - 20) , dq = \\int\_{0}^{20} (40 - q) , dq$$CS = \[40q - \\frac{q^2}{2}\]\_{0}^{20} = (40 \\times 20 - \\frac{20^2}{2}) - (0 - 0) = 800 - 200 = 600$ dollars [16](#page=16) [17](#page=17) [18](#page=18). Producer Surplus: $PS = \\int\_{0}^{20} (20 - q) , dq$$PS = \[20q - \\frac{q^2}{2}\]\_{0}^{20} = (20 \\times 20 - \\frac{20^2}{2}) - (0 - 0) = 400 - 200 = 200$ dollars [16](#page=16) [19](#page=19). ### 2.2 Gini coefficient The Gini coefficient is a measure of income or wealth inequality within a population, ranging from 0 (perfect equality) to 1 (perfect inequality). It is derived from the Lorenz curve, which plots the cumulative proportion of total income or wealth against the cumulative proportion of the population [20](#page=20). The Gini coefficient can be calculated using the definite integral of the Lorenz curve, denoted as $L(x)$, where $x$ represents the cumulative proportion of the population [20](#page=20). The formula for the Gini coefficient is: $$Gini = 2 \\left( \\frac{1}{2} - \\int\_{0}^{1} L(x) , dx \\right)$$ This can be simplified to: $$Gini = 1 - 2 \\int\_{0}^{1} L(x) , dx$$ > **Tip:** The term $\\int\_{0}^{1} L(x) , dx$ represents the area under the Lorenz curve. The area between the line of perfect equality (a 45-degree line) and the Lorenz curve is related to this integral. The Gini coefficient quantifies this area relative to the total area under the line of perfect equality. * * * # Improper integrals: definition, types, and convergence This section introduces improper integrals, which extend the concept of definite integration to cases involving infinite limits of integration or discontinuities within the integration interval. ### 3.1 Introduction to improper integrals Improper integrals are used in various applications, including statistics, to calculate probabilities involving continuous random variables with probability density functions. For example, the probability that a continuous random variable X lies between 'a' and 'b' is given by the integral of its probability density function p(x) from a to b: $P(a < X < b) = \\int\_{a}^{b} p(x) dx$. Similarly, probabilities involving infinite ranges are expressed as improper integrals, such as $P(X > 0) = \\int\_{0}^{+\\infty} p(x) dx$ and the total probability is $P(-\\infty < X < +\\infty) = 1 = \\int\_{-\\infty}^{+\\infty} p(x) dx$ [22](#page=22). ### 3.2 Type I improper integrals: infinite limits Type I improper integrals involve integration over an infinite interval. There are three definitions based on the position of the infinite limit(s) [23](#page=23): * **Integral from a to positive infinity:**$$ \\int\_{a}^{+\\infty} f(x) dx = \\lim\_{t \\to +\\infty} \\int\_{a}^{t} f(x) dx $$ where $a \\in \\mathbb{R}$ [23](#page=23). * **Integral from negative infinity to b:**$$ \\int\_{-\\infty}^{b} f(x) dx = \\lim\_{t \\to -\\infty} \\int\_{b}^{t} f(x) dx $$ where $b \\in \\mathbb{R}$ [23](#page=23). * **Integral from negative infinity to positive infinity:**$$ \\int\_{-\\infty}^{+\\infty} f(x) dx = \\lim\_{t \\to -\\infty} \\int\_{c}^{t} f(x) dx + \\lim\_{s \\to +\\infty} \\int\_{c}^{s} f(x) dx $$ where $c \\in \\mathbb{R}$. This integral converges if and only if both of the separate integrals converge [23](#page=23). #### 3.2.1 Convergence of Type I integrals An improper integral of Type I is considered **convergent** if the limit exists and is a finite real number. If the limit is infinite or does not exist, the integral is **divergent** [23](#page=23). > **Tip:** When evaluating improper integrals of Type I, the key is to replace the infinite limit with a variable and then evaluate the definite integral. The convergence or divergence is determined by the limit of this expression as the variable approaches infinity. #### 3.2.2 Example of Type I convergence Consider the integral $\\int\_{a}^{+\\infty} \\frac{1}{x^2} dx$ [24](#page=24). We evaluate this as: $$ \\int\_{a}^{+\\infty} \\frac{1}{x^2} dx = \\lim\_{t \\to +\\infty} \\int\_{a}^{t} \\frac{1}{x^2} dx $$$$ = \\lim\_{t \\to +\\infty} \\left\[ -\\frac{1}{x} \\right\]\_{a}^{t} $$$$ = \\lim{t \\to +\\infty} \\left( -\\frac{1}{t} + \\frac{1}{a} \\right) $$$$ = 0 + \\frac{1}{a} = \\frac{1}{a} $$ Since $a \\in \\mathbb{R}$ and the limit results in a finite value $\\frac{1}{a}$, the integral is **convergent** [24](#page=24). ### 3.3 Type II improper integrals: discontinuities Type II improper integrals involve integration over an interval where the integrand has an infinite discontinuity. These discontinuities can occur at one or both endpoints of the interval, or at an interior point [25](#page=25). #### 3.3.1 Definitions for Type II integrals The definitions for Type II improper integrals depend on the location of the discontinuity: * **f is discontinuous at the lower limit 'a' and continuous on (a, b:**$$ \\int\_{a}^{b} f(x) dx = \\lim\_{t \\downarrow a} \\int\_{t}^{b} f(x) dx $$ The notation $t \\downarrow a$ signifies that $t$ approaches $a$ from the right (values greater than $a$). * **f is discontinuous at the upper limit 'b' and continuous on \[a, b\[:**$$ \\int\_{a}^{b} f(x) dx = \\lim\_{t \\uparrow b} \\int\_{a}^{t} f(x) dx $$ The notation $t \\uparrow b$ signifies that $t$ approaches $b$ from the left (values less than $b$). * **f is discontinuous at an interior point 'c' within (a, b):**$$ \\int\_{a}^{b} f(x) dx = \\lim\_{t \\uparrow c} \\int\_{a}^{t} f(x) dx + \\lim\_{t \\downarrow c} \\int\_{t}^{b} f(x) dx $$ This integral converges if and only if both of the separate integrals converge. > **Remark:** If the integrand $f(x)$ is discontinuous at multiple points within the interval $\[a, b\]$, the interval should be split into sub-intervals, each containing a single point of discontinuity, and the definitions above should be applied to each sub-interval separately [25](#page=25). #### 3.3.2 Example of Type II divergence Consider the integral $\\int\_{2}^{4} \\frac{1}{(x-2)^{3/2}} dx$. Here, the integrand has a discontinuity at $x=2$. We apply the definition for a discontinuity at the lower limit [26](#page=26): $$ \\int\_{2}^{4} \\frac{1}{(x-2)^{3/2}} dx = \\lim\_{t \\downarrow 2} \\int\_{t}^{4} \\frac{1}{(x-2)^{3/2}} dx $$$$ = \\lim\_{t \\downarrow 2} \\left\[ -2(x-2)^{-1/2} \\right\]\_{t}^{4} $$$$ = \\lim{t \\downarrow 2} \\left( -2(4-2)^{-1/2} - (-2(t-2)^{-1/2}) \\right) $$$$ = \\lim\_{t \\downarrow 2} \\left( -2 ^{-1/2} + \\frac{2}{(t-2)^{1/2}} \\right) $$ [2](#page=2). $$ = -2 \\cdot \\frac{1}{\\sqrt{2}} + \\lim\_{t \\downarrow 2} \\frac{2}{\\sqrt{t-2}} $$$$ = -\\sqrt{2} + (+\\infty) = +\\infty $$ Since the limit is infinite, the integral is **divergent** [26](#page=26). ### 3.4 Exercise: $\\int\_{0}^{1} \\ln x , dx$ **1\. Why is $I = \\int\_{0}^{1} \\ln x , dx$ an improper integral?** The integrand $\\ln x$ has a discontinuity at $x=0$, which is the lower limit of integration. Therefore, it is a Type II improper integral [27](#page=27). **2\. Is $I$ convergent or divergent?** To determine convergence, we evaluate the integral using the definition for a discontinuity at the lower limit: $$ I = \\int\_{0}^{1} \\ln x , dx = \\lim\_{t \\downarrow 0} \\int\_{t}^{1} \\ln x , dx $$ We use integration by parts to evaluate $\\int \\ln x , dx$: let $u = \\ln x$ and $dv = dx$. Then $du = \\frac{1}{x} dx$ and $v = x$. $$ \\int \\ln x , dx = x \\ln x - \\int x \\cdot \\frac{1}{x} dx = x \\ln x - \\int 1 , dx = x \\ln x - x $$ Now, we apply the limits: $$ \\lim\_{t \\downarrow 0} \[x \\ln x - x\]\_{t}^{1} = \\lim{t \\downarrow 0} \\left( (1 \\ln 1 - 1) - (t \\ln t - t) \\right) $$$$ = \\lim\_{t \\downarrow 0} \\left( (0 - 1) - (t \\ln t - t) \\right) $$$$ = -1 - \\lim\_{t \\downarrow 0} (t \\ln t) + \\lim\_{t \\downarrow 0} t $$ We need to evaluate $\\lim\_{t \\downarrow 0} t \\ln t$. This is an indeterminate form of type $0 \\cdot (-\\infty)$. We can rewrite it as $\\lim\_{t \\downarrow 0} \\frac{\\ln t}{1/t}$, which is of type $\\frac{-\\infty}{\\infty}$. Using L'Hôpital's Rule: $$ \\lim\_{t \\downarrow 0} \\frac{\\ln t}{1/t} = \\lim\_{t \\downarrow 0} \\frac{1/t}{-1/t^2} = \\lim\_{t \\downarrow 0} -t = 0 $$ So, $\\lim\_{t \\downarrow 0} t \\ln t = 0$. Substituting this back into our expression for $I$: $$ I = -1 - 0 + 0 = -1 $$ Since the limit results in a finite number (-1), the integral $I = \\int\_{0}^{1} \\ln x , dx$ is **convergent** [28](#page=28). * * * ## Common mistakes to avoid * Review all topics thoroughly before exams * Pay attention to formulas and key definitions * Practice with examples provided in each section * Don't memorize without understanding the underlying concepts

| Term | Definition | |------|------------| | Definite integral | The definite integral of a continuous function $f$ on the interval $[a,b]$, denoted as $\int_a^b f(x) dx$, represents the net signed area between the function's graph and the x-axis from $a$ to $b$. It is calculated using the antiderivative $F$ of $f$ as $F(b) - F(a)$. | | Primitive function | A primitive function $F(x)$ of a function $f(x)$ is any function whose derivative is $f(x)$, i.e., $F'(x) = f(x)$. When calculating definite integrals, any primitive function can be used, and the constant of integration can be omitted. | | Area under a curve | When a function $f(x)$ is positive on an interval $[a,b]$, the definite integral $\int_a^b f(x) dx$ geometrically represents the area of the region bounded by the graph of $f(x)$, the x-axis, and the vertical lines $x=a$ and $x=b$. | | Negative function area | If a function $f(x)$ is negative on an interval $[a,b]$, the definite integral $\int_a^b f(x) dx$ will be negative. The geometric area of the region between the graph and the x-axis is then given by $-\int_a^b f(x) dx$. | | Consumer surplus | Consumer surplus (CS) is an economic measure representing the benefit consumers receive when they pay less for a product than they are willing to pay. It is calculated as the integral of the demand function minus the market price, from zero to the quantity purchased. | | Producer surplus | Producer surplus (PS) is an economic measure representing the benefit producers receive when they sell a product for more than they are willing to sell it for. It is calculated as the integral of the market price minus the supply function, from zero to the quantity sold. | | Gini coefficient | The Gini coefficient is a measure of statistical dispersion intended to represent the income or wealth distribution of a nation's residents, and is the most commonly used measure of inequality. It is calculated using the Lorenz curve and can be expressed using an integral. | | Improper integral | An improper integral is a definite integral where the interval of integration is infinite or the integrand has a discontinuity within the interval. These integrals are evaluated using limits. | | Type I improper integral | A Type I improper integral involves integration over an infinite interval, either from a finite number to positive infinity ($\int_a^\infty f(x) dx$), from negative infinity to a finite number ($\int_{-\infty}^b f(x) dx$), or over the entire real line ($\int_{-\infty}^\infty f(x) dx$). | | Type II improper integral | A Type II improper integral involves an integrand that is discontinuous at one or more points within the interval of integration. The integral is then defined as a limit of definite integrals over subintervals. | | Convergence of an integral | An improper integral is said to converge if its limiting value exists and is a finite real number. If the limit does not exist or is infinite, the integral is said to diverge. | | Divergence of an integral | An improper integral diverges if its limiting value does not exist or is infinite. This means that the area represented by the integral is either unbounded or does not approach a specific finite value. |