Riemann sums and definite integral - Calculus AB AP Study Notes

Riemann Sums are a method of approximating the area under a curve by dividing the region into rectangles and summing their areas. This foundational concept leads directly to the definite integral.

Key Definitions:

Partition: A division of interval [a,b] into n subintervals, creating points: a = x₀ < x₁ < x₂ < ... < xₙ = b

Δx (Width): For regular partitions, Δx = (b-a)/n, where n is the number of rectangles

Sample Point (x*ᵢ): A point chosen within each subinterval to determine rectangle height

Types of Riemann Sums:

• Left Riemann Sum: Uses left endpoint of each subinterval: Σf(xᵢ₋₁)Δx from i=1 to n
• Right Riemann Sum: Uses right endpoint: Σf(xᵢ)Δx from i=1 to n
• Midpoint Riemann Sum: Uses midpoint of each subinterval: Σf((xᵢ₋₁+xᵢ)/2)Δx

The Definite Integral is defined as the limit of Riemann sums as n→∞:

∫ᵇₐ f(x)dx = lim(n→∞) Σf(x*ᵢ)Δx

Fundamental Properties:

• ∫ᵇₐ f(x)dx represents the signed area between f(x) and the x-axis
• Area below x-axis contributes negatively
• ∫ᵃₐ f(x)dx = 0 (same limits)
• ∫ᵇₐ f(x)dx = -∫ᵃᵦ f(x)dx (reversing limits changes sign)

Memory Aid - RISE: Rectangles Increase, Sums Equal integral. As rectangles multiply, approximation accuracy rises!

Notation: The integral symbol ∫ is an elongated S (for "sum"), reminding us that integration is sophisticated summation.

The Intuition Behind Riemann Sums

Imagine measuring irregular coastline length. You can't use a straight ruler effectively, so you approximate using small straight segments. More segments = better approximation. Riemann sums work identically for area!

Real-World Application 1: Distance from Velocity

A car's speedometer shows varying velocities. To find total distance traveled, we need to "accumulate" velocity over time. If velocity v(t) = 2t meters/second, how far does it travel from t=0 to t=4 seconds?

Using 4 rectangles (Right Riemann):

• Δt = (4-0)/4 = 1 second
• Heights: v(1)=2, v(2)=4, v(3)=6, v(4)=8
• Distance ≈ 1(2+4+6+8) = 20 meters

As rectangles increase, we approach the exact answer: ∫₀⁴ 2t dt = 16 meters

Real-World Application 2: Water Flow

A pipe's flow rate varies: r(t) = 3 + sin(t) liters/minute. Total water delivered from t=0 to t=π minutes equals ∫₀^π (3+sin(t))dt. Each thin time slice contributes r(t)·Δt liters.

Why Three Types?

Left Riemann Sum: Underestimates increasing functions, overestimates decreasing ones Right Riemann Sum: Opposite behavior Midpoint: Generally most accurate for smooth functions—balances overestimation and underestimation

The Analogy of Pixelation: Think of Riemann sums as pixelating a smooth image. Low resolution (few rectangles) = blocky approximation. High resolution (many rectangles) = nearly perfect representation. The definite integral is infinite resolution!

Key Insight: Riemann sums transform instantaneous rates into accumulated quantities—the bridge between differential and integral calculus.