Business Calculus: Antiderivatives, Definite Integrals, and Applications

4.1: Anti-differentiation

Definition and Basic Concepts

Anti-differentiation, also known as finding the antiderivative or indefinite integration, is the process of determining a function whose derivative is the given function. In other words, if $\frac{d}{dx}F(x) = f(x)$, then $F(x)$ is an antiderivative of $f(x)$.

• Antiderivative: If $f(x)$ is a function, then any function $F(x)$ such that $F'(x) = f(x)$ is called an antiderivative of $f(x)$.

• General Antiderivative: The set of all antiderivatives of $f(x)$ is $F(x) + C$, where $C$ is an arbitrary constant, called the constant of integration.

Example: If $f(x) = 2x$, then $F(x) = x^2$ is an antiderivative, since $\frac{d}{dx}x^2 = 2x$.

Formula:

• $\int f(x)dx = F(x) + C$

4.2: Rules of Antiderivatives

Basic Integration Rules

Several standard rules are used to find antiderivatives of common functions:

• Power Rule: $\int x^n dx = \frac{x^{n+1}}{n+1} + C$, where $n \neq -1$

• Exponential Rule: $\int e^x dx = e^x + C$

• Natural Logarithm Rule: $\int \frac{1}{x} dx = \ln|x| + C$, $x > 0$

• Constant Multiple Rule: $\int k f(x) dx = k \int f(x) dx$

• Sum Rule: $\int [f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx$

Example: $\int (2x^3 + 5e^x) dx = 2 \int x^3 dx + 5 \int e^x dx = 2 \cdot \frac{x^4}{4} + 5e^x + C = \frac{x^4}{2} + 5e^x + C$

Applications: Population Modeling

Exponential Growth Model

In business calculus, exponential functions are often used to model population growth. For example, the rate of change of the population of Phoenix, Arizona, can be modeled by an exponential function $P'(t)$, where $t$ is the number of years since 1960 and $P'(t)$ is in thousands of people per year.

• Given $P'(t)$ and an initial population, integrate $P'(t)$ to find $P(t)$, the population function.

• Use initial conditions to solve for the constant of integration.

Example: If $P'(t)$ is given and $P(20) = 790,000$, integrate $P'(t)$ and use the initial value to find $P(t)$.

4.2: Antiderivatives as Areas

Summation and Sigma Notation

Summation notation (sigma notation) is used to represent the sum of a sequence of terms. It is essential for understanding the connection between antiderivatives and areas under curves.

• General Form: $\sum_{k=m}^n a_k = a_m + a_{m+1} + \cdots + a_n$

• Example: $\sum_{k=1}^4 \frac{2}{k} = 2/1 + 2/2 + 2/3 + 2/4$

Area Under a Curve

The area under a curve can be approximated by dividing the region into rectangles (Riemann sums). The more rectangles used, the better the approximation.

• Divide the interval $[a, b]$ into $n$ subintervals of equal width: $\Delta x = \frac{b-a}{n}$

• Construct rectangles above the subintervals, touching the curve at a chosen point (left, right, or midpoint).

• Sum the areas of the rectangles to approximate the total area under the curve.

Example: Approximate the area under $f(x) = 600x - x^2$ over $[0, 600]$ by dividing into 6 or 12 subintervals.

4.3: Area and Definite Integrals

Definite Integral and Area

The definite integral of a function $f(x)$ from $a$ to $b$ gives the net area under the curve between $x = a$ and $x = b$.

• Definition: $\int_a^b f(x) dx = F(b) - F(a)$, where $F(x)$ is any antiderivative of $f(x)$.

Fundamental Theorem of Calculus: If $f(x)$ is continuous on $[a, b]$ and $F(x)$ is an antiderivative of $f(x)$, then:

• $\int_a^b f(x) dx = F(b) - F(a)$

This theorem connects differentiation and integration, allowing us to compute areas exactly.

Properties of Definite Integrals

• Additivity: $\int_a^c f(x) dx = \int_a^b f(x) dx + \int_b^c f(x) dx$ for $a < b < c$

• Piecewise Functions: For piecewise-defined functions, break the integral at the points where the definition changes.

Example Table: Piecewise Function Integration

Function	Interval	Integral
$f(x) = \begin{cases} 9, & x < 3 \\ x^2, & x \geq 3 \end{cases}$	$x$ from $-4$ to $5$	$\int_{-4}^3 9 dx + \int_3^5 x^2 dx$

Area Between Two Curves

The area of a region bounded by two curves $f(x)$ and $g(x)$ over $[a, b]$ (where $f(x) \geq g(x)$) is:

• $A = \int_a^b [f(x) - g(x)] dx$

Steps:

1. Find the points of intersection to determine $a$ and $b$.

2. Identify the top and bottom curves.

3. Integrate the difference $f(x) - g(x)$ from $a$ to $b$.

Example: Find the area between $f(x) = 2x + 1$ and $g(x) = x^2 + 1$ from $x = 0$ to $x = 2$.

4.4: Properties of Definite Integrals

Key Properties

• Linearity: $\int_a^b [cf(x) + dg(x)] dx = c\int_a^b f(x) dx + d\int_a^b g(x) dx$

• Reversal of Limits: $\int_a^b f(x) dx = -\int_b^a f(x) dx$

• Zero Width: $\int_a^a f(x) dx = 0$

Average Value of a Function

The average value of a continuous function $f(x)$ over $[a, b]$ is:

• $f_{\text{avg}} = \frac{1}{b-a} \int_a^b f(x) dx$

Example: The average value of $f(x) = x^2$ over $[0, 2]$ is $\frac{1}{2-0} \int_0^2 x^2 dx$.

4.5: Substitution Integration Technique

u-Substitution

Substitution is a method for evaluating integrals by changing variables to simplify the integrand.

• Let $u$ be a function of $x$ (e.g., the inner function in a composite function).

• Compute $du = u'(x) dx$.

• Rewrite the integral in terms of $u$ and $du$.

• Integrate with respect to $u$.

• Substitute back in terms of $x$.

Example: $\int 2x e^{x^2} dx$ Let $u = x^2$, $du = 2x dx$, so the integral becomes $\int e^u du = e^u + C = e^{x^2} + C$.

4.6: Integration by Parts Technique

Integration by Parts Formula

Integration by parts is used for products of functions and is based on the product rule for differentiation.

• Formula: $\int u dv = uv - \int v du$

• Choose $u$ and $dv$ such that $u$ can be differentiated and $dv$ can be integrated.

• Differentiate $u$ to get $du$, integrate $dv$ to get $v$.

• Apply the formula and simplify.

Example: $\int x e^x dx$ Let $u = x$, $dv = e^x dx$; then $du = dx$, $v = e^x$. So, $\int x e^x dx = x e^x - \int e^x dx = x e^x - e^x + C$.

4.7: Numerical Integration

Approximate Integration Methods

When an integral cannot be evaluated exactly, numerical methods such as the Trapezoidal Rule or Simpson's Rule are used to approximate the value.

• Trapezoidal Rule: Approximates the area under a curve by dividing it into trapezoids.

• Simpson's Rule: Uses parabolic arcs instead of straight lines for better accuracy.

Example Table: Comparison of Numerical Methods

Method	Approach	Accuracy
Left/Right Riemann Sum	Rectangles	Low (depends on function)
Trapezoidal Rule	Trapezoids	Moderate
Simpson's Rule	Parabolic arcs	High (for smooth functions)

Appendix: Common Antiderivatives

Table of Indefinite Integrals

Integral	Result	Conditions
$\int x^n dx$	$\frac{x^{n+1}}{n+1} + C$	$n \neq -1$
$\int \frac{1}{x} dx$	$\ln|x| + C$	$x > 0$
$\int e^x dx$	$e^x + C$
$\int a^x dx$	$\frac{a^x}{\ln a} + C$	$a > 0, a \neq 1$
$\int \sin x dx$	$-\cos x + C$
$\int \cos x dx$	$\sin x + C$
$\int \frac{1}{x^2 + a^2} dx$	$\frac{1}{a} \arctan\left(\frac{x}{a}\right) + C$	$a > 0$

Additional info: Some examples and tables have been expanded for clarity and completeness. The notes cover the core topics of antiderivatives, definite integrals, area under curves, properties of integrals, substitution, integration by parts, and numerical integration, all of which are central to a Business Calculus course.