8.2 Definite Integration and Numerical Methods

Definite integration is a powerful tool in calculus, connecting antiderivatives to areas under curves. The Fundamental Theorem of Calculus provides a method for calculating definite integrals, while numerical techniques like the Trapezoidal Rule offer approximations.

Definite integrals have numerous real-world applications, from finding areas between curves to calculating volumes of solids. Improper integrals extend these concepts to handle infinite limits and discontinuous functions, broadening the scope of integration techniques.

Definite Integration

Fundamental Theorem of Calculus

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• Establishes a connection between definite integrals and antiderivatives (indefinite integrals)
  • If $F(x)$ is an antiderivative of $f(x)$, then $\int_a^b f(x) dx = F(b) - F(a)$ (evaluates the definite integral)
• Steps to calculate a definite integral using the FTC:
  1. Find an antiderivative $F(x)$ of the integrand $f(x)$ (using integration techniques)
  2. Evaluate $F(x)$ at the upper limit $b$ and lower limit $a$ of integration
  3. Subtract the value of $F(a)$ from $F(b)$ to obtain the definite integral
• Geometrically represents the signed area between the curve $y = f(x)$ and the x-axis over the interval $[a, b]$ (positive area above x-axis, negative area below)

Numerical methods for integration

• Approximate the value of a definite integral numerically
• Trapezoidal Rule approximates the area under a curve using trapezoids
  • Divides the interval $[a, b]$ into $n$ equal subintervals of width $h = \frac{b-a}{n}$
  • Approximates the definite integral using the formula: $\int_a^b f(x) dx \approx \frac{h}{2}[f(x_0) + 2f(x_1) + 2f(x_2) + \ldots + 2f(x_{n-1}) + f(x_n)]$, where $x_i = a + ih$
• Simpson's Rule approximates the area using quadratic polynomials (parabolic arcs)
  • Divides the interval $[a, b]$ into an even number of subintervals
  • Approximates the definite integral using the formula: $\int_a^b f(x) dx \approx \frac{h}{3}[f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \ldots + 2f(x_{n-2}) + 4f(x_{n-1}) + f(x_n)]$, where $h = \frac{b-a}{n}$ and $x_i = a + ih$
• Accuracy of the approximation improves as the number of subintervals increases (smaller $h$)

Applications and Improper Integrals

Applications of definite integrals

• Area between curves $y = f(x)$ and $y = g(x)$ over $[a, b]$: $\int_a^b [f(x) - g(x)] dx$ (subtracts lower curve from upper curve)
• Volumes of solids of revolution:
  • Disk method: $V = \int_a^b \pi [f(x)]^2 dx$ (revolves region between $y = f(x)$ and x-axis about x-axis)
  • Shell method: $V = \int_a^b 2\pi x f(x) dx$ (revolves region between $y = f(x)$ and x-axis about y-axis)
• Arc length of a curve $y = f(x)$ over $[a, b]$: $L = \int_a^b \sqrt{1 + [f'(x)]^2} dx$ (integrates the square root of 1 plus the squared derivative)

Improper integrals

• Involve infinite limits of integration or discontinuous integrands
• Infinite limits:
  • $\int_a^{\infty} f(x) dx = \lim_{b \to \infty} \int_a^b f(x) dx$ (upper limit approaches infinity)
  • $\int_{-\infty}^b f(x) dx = \lim_{a \to -\infty} \int_a^b f(x) dx$ (lower limit approaches negative infinity)
• Discontinuous integrands:
  • If $f(x)$ has a discontinuity at $c$ in $[a, b]$, then $\int_a^b f(x) dx = \int_a^c f(x) dx + \int_c^b f(x) dx$ (splits integral at discontinuity)
• Improper integrals converge if the limits exist and are finite, otherwise they diverge (fail to converge)