Area Estimates

Learn left riemann sums, right riemann sums.

Lesson 1 of 32 parts Practice Flashcards

1/2

Left Riemann Sums

Objective: Explain left riemann sums, apply the stated method, and check its conditions.

A Riemann sum multiplies sample heights by subinterval widths and adds the resulting signed contributions. A left sum samples each subinterval’s left endpoint. For an increasing function, a left sum underestimates its definite integral on an equal partition. An estimate’s direction depends on the function’s behavior.

Formulas

$L=Σ f(xᵢ₋₁)Δxᵢ$

Worked examples

Try itEstimate ∫₀^7 x dx with 7 unit-width rectangles using left endpoints.Show answer

Step 1: Width is 1; sample heights are 0,1,…,6. Step 2: Add heights using the arithmetic-sum formula. Step 3: The left estimate is 21.

Swipe sideways to move between sections

Left Riemann Sums

Objective: Explain left riemann sums, apply the stated method, and check its conditions.

Formulas

$L=Σ f(xᵢ₋₁)Δxᵢ$

Worked examples

Try itEstimate ∫₀^7 x dx with 7 unit-width rectangles using left endpoints.Show answer

Step 1: Width is 1; sample heights are 0,1,…,6. Step 2: Add heights using the arithmetic-sum formula. Step 3: The left estimate is 21.

Right Riemann Sums

Objective: Explain right riemann sums, apply the stated method, and check its conditions.

A right Riemann sum uses each subinterval’s right endpoint as the sample. Widths must correspond to the partition, and endpoints must not add an extra rectangle. For an increasing function, right samples overestimate the definite integral. A partition with n intervals has n rectangles and n+1 boundary points.

Formulas

$R=Σ f(xᵢ)Δxᵢ$

Worked examples

Try itEstimate ∫₀^7 x dx with 7 unit-width rectangles using right endpoints.Show answer

Step 1: Heights are 1,2,…,7. Step 2: Multiply the sum by width 1. Step 3: The right estimate is 28.

You finished the reading!

Lock it in with a few practice questions, then move on to the next lesson.

Practice Next lesson

Next lessonDefinite Integrals