Chapter 9 Integration Review

Definite Integrals

  • Given a function y = f(x), the definite integral from a to b represents the signed area between the function's graph and the x-axis.
  • If f(x) is positive, the area is above the x-axis and considered positive.
  • If f(x) is negative, the area is below the x-axis and considered negative.
  • The definite integral is the sum of the positive and negative areas.
  • \int_{a}^{b} f(x) dx = \text{signed area between } f(x) \text{ and the x-axis from } a \text{ to } b

Calculating Definite Integrals via Area

  • If the area can be calculated geometrically (e.g., triangles, trapezoids), the definite integral can be found without calculus.
  • Example: Integral from 0 to 6 of a triangle function with height 1 at x=3 can be calculated as the area of the triangle, which is \frac{1}{2} * 6 * 1 = 3 .

Finding x Value for a Given Integral Value

  • Problem: Find x s.t. \int_{0}^{x} f(t) dt = 2.
  • If x is between 3 and 6, the left side of the area is a triangle and the right side of the area is a triploid.
  • x = 6 - \sqrt{6}

Fundamental Theorem of Calculus for Definite Integrals

  • If finding area directly is not possible, this theorem is used to calculate integrals.
  • \int_{a}^{b} f(x) dx = g(b) - g(a), where g'(x) = f(x) (g is antiderivative of f).
  • A special choice for the antiderivative is g(x) = \int_{c}^{x} f(t) dt, where c is any fixed number.
  • The derivative of this integral function is f(x).
  • If F(x) = \int_{0}^{x} f(t) dt, then F'(x) = f(x).

Indefinite Integrals

  • Represents the general antiderivative of a function.
  • Includes all possible antiderivatives, differing by a constant.
  • \int f(x) dx = F(x) + C, where F'(x) = f(x) and C is the integration constant.
  • Example: \int 3 dx = 3x + C
  • \int x dx = \frac{1}{2}x^2 + C

Power Rule for Indefinite Integrals

  • \int x^a dx = \frac{1}{a+1}x^{a+1} + C, for a \neq -1
  • Examples:
    • \int \sqrt{x} dx = \frac{2}{3}x^{\frac{3}{2}} + C
    • \int \frac{1}{\sqrt{x}} dx = 2\sqrt{x} + C

Fundamental Theorem of Calculus for Indefinite Integrals

  • \frac{d}{dx} \int f(x) dx = f(x)
  • \int g'(x) dx = g(x) + C