Integration and Its Applications in Business Calculus

Chapter 5 - Integration

5.1 Antiderivatives & Indefinite Integrals

The process of finding a function whose derivative is a given function is called antidifferentiation. The set of all antiderivatives of a function is called the indefinite integral.

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

• Indefinite Integral: The general form is $\int f(x)dx = F(x) + C$, where C is the constant of integration.

• Formulas:

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

  • $\int x^{-1} dx = \ln|x| + C$

  • $\int e^x dx = e^x + C$

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

  • $\int kf(x)dx = k\int f(x)dx$

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

• Example: $\int 2x dx = x^2 + C$

5.2 Integration by Substitution

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

• Steps:

  1. Let $u$ be a function inside the integrand.

  2. Compute $du$ and substitute into the integral.

  3. Integrate with respect to $u$.

  4. Substitute back in terms of $x$.

• Example: $\int (3x^2 + 4)^4 6x dx$ Let $u = 3x^2 + 4$, $du = 6x dx$, so $\int u^4 du = \frac{1}{5}u^5 + C = \frac{1}{5}(3x^2 + 4)^5 + C$

5.4 The Definite Integral

The definite integral computes the net area under a curve between two points. It is defined as the limit of Riemann sums as the number of subintervals approaches infinity.

• Geometric Interpretation: The area under the velocity-time graph gives the distance traveled.

• Riemann Sums: Approximating the area under a curve by summing areas of rectangles.

• Definite Integral: $\int_a^b f(x)dx$ is the exact area under $f(x)$ from $x=a$ to $x=b$.

• Properties:

  • $\int_a^a f(x)dx = 0$

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

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

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

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

5.5 The Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects differentiation and integration. If $F$ is any antiderivative of $f$ on $[a, b]$, then:

• $\int_a^b f(x)dx = F(b) - F(a)$

• Application: To evaluate a definite integral, find any antiderivative and subtract its value 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 $[a, b]$.

• Example: Find the area between $f(x) = \frac{1}{2}x + 3$ and $g(x) = -x^2 + 1$ from $x = -2$ to $x = 1$.

Income Distribution and the Lorenz Curve

The Lorenz curve is a graphical representation of income distribution. The Gini index measures income inequality and is calculated as:

• $\text{Gini Index} = 2\int_0^1 [x - f(x)] dx$

• A Gini Index of 0 means perfect equality; 1 means perfect inequality.

Area Between More Complex Curves

• For regions bounded by intersecting curves, split the area at intersection points and integrate accordingly.

Probability Density Functions

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

• $f(x) \geq 0$ for all $x$

• $\int_{-\infty}^{\infty} f(x)dx = 1$

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

Consumers' and Producers' Surplus

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

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

• Where $(\bar{x}, \bar{p})$ is the equilibrium point of demand $D(x)$ and supply $S(x)$.