Integration and Its Applications in Business Calculus

Chapter 5 - Integration

5.1 Antiderivatives & Indefinite Integrals

Antiderivatives are the reverse operation of differentiation. If a function F(x) has a derivative equal to f(x), then F(x) is called an antiderivative of f(x). The collection of all antiderivatives of a function is called the indefinite integral.

• Definition: If F'(x) = f(x), then F(x) is an antiderivative of f(x).

• Notation: The indefinite integral is written as $\int f(x)dx = F(x) + C$, where C is the constant of integration.

• Key Properties:

  • $\int x^n dx = \frac{1}{n+1}x^{n+1} + C$, for $n \neq -1$

  • $\int x^{-1} dx = \ln|x| + C$

  • $\int e^x dx = e^x + C$

  • $\int e^{kx} dx = \frac{1}{k}e^{kx} + C$

  • $\int kf(x)dx = k\int f(x)dx$

  • $\int [f(x) \pm g(x)] dx = \int f(x)dx \pm \int g(x)dx$

• Example: Find an antiderivative for $f(x) = 2x$. One possible answer is $F(x) = x^2$.

5.2 Integration by Substitution

Integration by substitution is a technique used to evaluate more complex integrals, essentially reversing the chain rule. If $u = f(x)$, then $du = f'(x)dx$.

• Steps:

  1. Choose a substitution $u = g(x)$ that simplifies the integral.

  2. Compute $du = g'(x)dx$.

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

  4. Integrate with respect to $u$.

  5. Substitute back in terms of $x$.

• Example: $\int (3x^2 + 4)^4 6x dx$. Let $u = 3x^2 + 4$, then $du = 6x dx$.

5.4 The Definite Integral

The definite integral computes the net area under a curve between two points. It is closely related to the concept of area and displacement in applications.

• Definition: $\int_a^b f(x)dx$ is the definite integral of f(x) from a to b.

• Interpretation: The definite integral represents the signed area between the curve and the x-axis from x = a to x = b.

• Properties:

  • $\int_a^a f(x)dx = 0$

  • $\int_b^a f(x)dx = -\int_a^b f(x)dx$

  • $\int_a^b kf(x)dx = k\int_a^b f(x)dx$

  • $\int_a^b [f(x) \pm g(x)] dx = \int_a^b f(x)dx \pm \int_a^b g(x)dx$

  • $\int_a^c f(x)dx = \int_a^b f(x)dx + \int_b^c f(x)dx$

• Riemann Sums: Approximates the area under a curve by summing areas of rectangles. As the number of rectangles increases, the approximation improves.

  • Left rectangles: Use the left endpoint for the height.

  • Right rectangles: Use the right endpoint for the height.

5.5 The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration. It states that if F is any antiderivative of f on [a, b], then:

• $\int_a^b f(x)dx = F(b) - F(a)$

• This theorem allows us to evaluate definite integrals using antiderivatives.

• Example: To evaluate $\int_1^2 (2x + 3e^x - 4/x)dx$, find an antiderivative and compute the difference at the endpoints.

Chapter 6 - Additional Integration Topics

6.1/6.2 Area Between Curves & Integration Applications

Area Between Curves

The area between two curves $y = f(x)$ and $y = g(x)$ from $x = a$ to $x = b$ is given by:

• $A = \int_a^b [f(x) - g(x)] dx$, where $f(x) \geq g(x)$ on [a, b].

• Signed area: Area above the x-axis minus area below the x-axis.

• Unsigned area: Always nonnegative, representing the total region between curves.

Applications in Business & Economics

Integration is used to analyze income distribution, continuous income streams, and consumer/producer surplus.

• Lorenz Curve: Graphs cumulative income distribution. The Gini Index measures income inequality as $Gini = 2\int_0^1 [x - f(x)] dx$.

• Continuous Income Stream: Total income over [a, b] is $\int_a^b f(t)dt$.

• Future Value: For continuous compounding at rate r, $FV = e^{rT}\int_0^T f(t)e^{-rt}dt$.

• Consumer Surplus: $CS = \int_0^{\bar{x}} [D(x) - \bar{p}] dx$

• Producer Surplus: $PS = \int_0^{\bar{x}} [\bar{p} - S(x)] dx$

Probability Density Functions

A probability density function (pdf) f(x) for a continuous random variable must satisfy:

• f(x) ≥ 0 for all x

• The total area under f(x) over all real numbers is 1

• The probability that x is in [c, d] is $\int_c^d f(x)dx$

Summary Table: Properties of Definite Integrals

Property	Formula
Zero interval	$\int_a^a f(x)dx = 0$
Reversal of limits	$\int_b^a f(x)dx = -\int_a^b f(x)dx$
Constant multiple	$\int_a^b kf(x)dx = k\int_a^b f(x)dx$
Sum/Difference	$\int_a^b [f(x) \pm g(x)] dx = \int_a^b f(x)dx \pm \int_a^b g(x)dx$
Additivity	$\int_a^c f(x)dx = \int_a^b f(x)dx + \int_b^c f(x)dx$