Area Under a Curve and Riemann Sums
Using rectangles to estimate the space bounded by a function.
Introduction
How do you find the area under a curved graph? Divide it into thin rectangles, sum their areas, and take the limit as the rectangles become infinitely thin. This Riemann sumapproach leads to the integral.
Prerequisite Connection
You understand summation notation and limits.
Today's Increment
We estimate areas using left, right, and midpoint Riemann sums.
Why This Matters
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!
Practice Quiz
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