Multiple Choice
For the following graph, write a Reimann sum using left endpoints to approximate the area under the curve over [0,6] with 6 subintervals.
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8. Definite Integrals
Riemann Sums
Multiple Choice
Express the integral as a limit of Riemann sums using right endpoints. Do not evaluate the limit.
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Verified step by step guidance1
Step 1: Understand the problem. The goal is to express the given integral ∫₀² (3x² + 1) dx as a limit of Riemann sums using right endpoints. Riemann sums approximate the area under a curve by dividing the interval into subintervals and summing up the areas of rectangles formed using function values at specific points (right endpoints in this case).
Step 2: Divide the interval [0, 2] into n subintervals of equal width. The width of each subinterval is Δx = (2 - 0)/n = 2/n.
Step 3: Identify the right endpoints of each subinterval. For the i-th subinterval, the right endpoint is xᵢ = 0 + iΔx = i(2/n).
Step 4: Evaluate the function at the right endpoints. Substitute xᵢ into the function f(x) = 3x² + 1 to get f(xᵢ) = 3(i(2/n))² + 1.
Step 5: Write the Riemann sum. The sum of the areas of the rectangles is Σᵢ₌₁ⁿ f(xᵢ)Δx = Σᵢ₌₁ⁿ [3(i(2/n))² + 1](2/n). Finally, express the integral as the limit of the Riemann sum as n → ∞: limₙ→∞ Σᵢ₌₁ⁿ [3(i(2/n))² + 1](2/n).
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