Integration and Its Applications Study Notes

Chapter 3: Integration and Its Applications

Definitions

• Antiderivative: A function F is called an antiderivative of f on an interval I if F'(x) = f(x) for all x in I. The process of recovering a function F(x) from its derivative f(x) is known as antidifferentiation.
  • Capital letters are used to denote antiderivatives: F for f, G for g, etc.

Examples on Finding Antiderivatives

Example 1: Finding Antiderivatives

• Find an antiderivative for the following functions:

  1. (a) f(x) = 2x
  2. (b) g(x) = cos(x)
  3. (c) h(x) = -2x + cos(x)

• Solutions:

  1. (a) F(x) = x²
  2. (b) G(x) = sin(x)
  3. (c) H(x) = -x² + sin(x)
  • Each answer can be verified by differentiating the antiderivative.

• Note: The antiderivative of f(x) = 2x is not limited to F(x) = x², but includes a collection of antiderivatives in the form x² + C, where C is any arbitrary constant.

  • Therefore, the general form of the antiderivative is F(x) + C.

Example 2: Finding a Particular Antiderivative

• Find an antiderivative of f(x) = sin(x) that satisfies F(0) = 3.
  • Since the derivative of -cos(x) is sin(x), the general antiderivative is F(x) = -cos(x) + C.
  • Applying the condition F(0) = 3:
  • F(0) = -cos(0) + C = -1 + C = 3.
  • Thus, C = 4, and so F(x) = -cos(x) + 4 is the required antiderivative.

Indefinite Integrals

• The collection of all antiderivatives of a function f is known as the indefinite integral of f with respect to x, denoted by:
  $\int f(x) \, dx$

  • Here, the integral sign indicates indefinite integral; f is the integrand; x is the variable of integration.

Integral Table

• Standard integrals can be summarized as:
  1. $\int x^n : dx = \frac{x^{n+1}}{n+1} + C, \ n \neq -1$
  2. $\int e^x : dx = e^x + C$
  3. $\int \sin x \, dx = - \cos x + C$
  4. $\int \cos x \, dx = \sin x + C$
  5. $\int \sec^2 x \, dx = \tan x + C$
  6. $\int \csc^2 x \, dx = -\cot x + C$
  7. $\int \sec x \tan x \, dx = \sec x + C$
  8. $\int \csc x \cot x \, dx = -\csc x + C$

Techniques of Integration

1. Integration by Substitution

• For algebraic and trigonometric substitutions:
  • Let $u = g(x)$, then $du = g'(x) \, dx$, and the integral becomes:
    $\int f(g(x)) g'(x) \, dx = \int f(u) : du$

2. Integration by Parts

• For the product of two functions:
  • $\int u \, dv = uv - \int v \, du$
  • Use ILATE order for choosing u:
  • Inverse Trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential Functions.

Improper Integrals

1. Type I: Integrals with limits of infinity:
   $\int<em>a^\infty f(x) \, dx = \lim</em>{b \to \infty} \int_a^b f(x) \, dx$
2. Type II: Integrals with discontinuities within limits:
   $\int<em>a^b f(x) \, dx = \lim</em>{c \to a} \int<em>a^c f(x) \, dx + \lim</em>{d \to b} \int_d^b f(x) \, dx$

Applications of Definite Integrals

• Use definite integrals to find:
  1. Area Under Curves: Compute the area using antiderivatives and the Fundamental Theorem of Calculus.
  2. For regions bounded between curves:
  • Subtract the paths of the curves from one another across their limits.
  1. Volume of Solids of Revolution: Generated by rotating a function about an axis (disk/washer methods).

Examples of Applications

• Area between curves and total area:

  • Use the negative areas' absolute values while calculating areas to ensure no cancellation occurs.

• Surface Area of Revolution: Computed using integral formulas specific for revolving curves around axes (x-axis/y-axis).

• Length of a Plane Curve: Formula takes into account the function derivatives, and whether it is parametric or defined directly as a function in x or y, respectively.

Closing Summary

• Integration connects areas, volumes, and solving differential equations. Mastery of substitution, parts, and understanding integral tables is essential.

Exercises

• Check various integrals, find total areas using definite integrals and apply to problems like areas under curves and volumes of revolution.