Numerical integration - Calculus AB AP Study Notes

Numerical integration approximates the definite integral ∫ₐᵇ f(x)dx when an antiderivative cannot be found analytically or when working with discrete data points. This is essential for calculating accumulation from rate functions.

Key Methods:

Left Riemann Sum: L_n = Δx[f(x₀) + f(x₁) + ... + f(x_{n-1})] where Δx = (b-a)/n. Uses left endpoints of subintervals, typically underestimates for increasing functions.

Right Riemann Sum: R_n = Δx[f(x₁) + f(x₂) + ... + f(xₙ)]. Uses right endpoints, typically overestimates for increasing functions.

Midpoint Rule: M_n = Δx[f(m₁) + f(m₂) + ... + f(mₙ)] where mᵢ is the midpoint of each subinterval. Generally more accurate than left/right sums.

Trapezoidal Rule: T_n = (Δx/2)[f(x₀) + 2f(x₁) + 2f(x₂) + ... + 2f(x_{n-1}) + f(xₙ)]. Averages left and right sums, forming trapezoids rather than rectangles. Note the pattern: first and last terms appear once, interior terms are doubled.

Memory Aid - "LRMT": Left is Lower (for increasing), Right is Raised, Midpoint is More accurate, Trapezoidal Takes the average.

Error Behavior: For concave up functions, L_n and M_n underestimate while R_n and T_n overestimate. This reverses for concave down functions. Increasing n always improves accuracy by reducing Δx.

Numerical integration solves practical problems where exact integration is impossible or impractical.

Real-World Application - Environmental Science: Consider measuring pollution levels in a river. Sensors collect discrete temperature readings every hour rather than continuous data. To find total heat energy accumulated over 24 hours, we cannot integrate a formula—we must use trapezoidal rule on the data points. Each trapezoid represents the average temperature between consecutive hours multiplied by one hour.

Engineering Example - Velocity Data: A car's speedometer records velocity every 5 seconds during acceleration testing. To determine distance traveled, we need ∫v(t)dt, but we only have data points, not a velocity function. The midpoint rule estimates distance by assuming constant velocity at each interval's midpoint—like approximating your journey by checking speed at the middle of each city block.

Medical Application: Heart rate monitors produce discrete pulse readings. Calculating total blood pumped requires integrating heart rate over time. With 60 readings per minute, left Riemann sums provide lower bounds (conservative estimates) for blood volume calculations.

Visual Analogy: Imagine calculating a pond's area from satellite images. Rectangles (Riemann sums) overlay the pond using grid squares—left/right positioning matters. Trapezoids fit the curved shoreline better by connecting points with straight lines. Midpoints sample the center of each grid square, often capturing the pond's shape more accurately. More subdivisions (larger n) mean finer grids and better approximations, just as higher-resolution images reveal more detail.