Lesson 22.3

Area Under a Curve and Riemann Sums

Using rectangles to estimate the space bounded by a function.

Introduction

Using rectangles to estimate the space bounded by a function.

Past Knowledge

You understand summation notation and limits.

Today's Goal

We estimate areas using left, right, and midpoint Riemann sums.

Future Success

This is the conceptual foundation for integrals—the other half of calculus!

Key Concepts

Rectangle Width

Where [a,b] is the interval and n is the number of rectangles.

Left Riemann Sum

Use left endpoint of each subinterval for height.

Right Riemann Sum

Use right endpoint of each subinterval for height.

The Definite Integral

Worked Examples

Example 1: Left Sum Setup (Basic)

Estimate area under from x = 0 to x = 2 using n = 4 left rectangles.

Left endpoints: 0, 0.5, 1, 1.5

Heights: 0, 0.25, 1, 2.25

L₄ = 0.5(0 + 0.25 + 1 + 2.25) = 1.75

Example 2: Right Sum (Intermediate)

Same problem with right rectangles.

Right endpoints: 0.5, 1, 1.5, 2

Heights: 0.25, 1, 2.25, 4

R₄ = 0.5(0.25 + 1 + 2.25 + 4) = 3.75

Example 3: Exact Area (Advanced)

The exact area is between L₄ and R₄.

Actual:

1.75 < 2.67 < 3.75 ✓

Common Pitfalls

Wrong number of terms

n rectangles means n heights to sum. Count carefully!

Confusing left vs right

Left starts at a, right ends at b. Draw a picture!

Real-World Application

Total Distance from Velocity

If you know velocity v(t), the area under the velocity curve gives total distance traveled. Car trip computers use Riemann sums to estimate distance from speedometer readings!