Identifying definite integrals as limits of sums Consider the following limits of Riemann sums for a function ƒ on [a,b]. Identify ƒ and express the limit as a definite integral.
n
lim ∑ 𝓍*ₖ (ln 𝓍*ₖ) ∆𝓍ₖ on [1,2]
∆ → 0 k=1
8. Definite Integrals
Riemann Sums
5:33 minutesProblem 5.1.61
Textbook Question
{Use of Tech} Sigma notation for Riemann sums Use sigma notation to write the following Riemann sums. Then evaluate each Riemann sum using Theorem 5.1 or a calculator.
The right Riemann sum for ƒ(𝓍)) = x + 1 on [0, 4] with n = 50.
Verified step by step guidance1
Step 1: Understand the problem. We are tasked with writing the right Riemann sum for the function ƒ(x) = x + 1 on the interval [0, 4] using sigma notation, with n = 50 subintervals. Then, we will evaluate the sum using Theorem 5.1 or a calculator.
Step 2: Divide the interval [0, 4] into n = 50 subintervals. The width of each subinterval, denoted as Δx, is calculated as Δx = (b - a) / n, where a = 0 and b = 4. Substitute the values to find Δx.
Step 3: Identify the right endpoints of each subinterval. The right endpoint of the i-th subinterval is given by xᵢ = a + iΔx, where i ranges from 1 to n. Substitute the values of a and Δx to express xᵢ in terms of i.
Step 4: Write the Riemann sum in sigma notation. The right Riemann sum is expressed as Σ (from i = 1 to n) of ƒ(xᵢ)Δx. Substitute ƒ(xᵢ) = xᵢ + 1 and the expression for xᵢ into the sigma notation.
Step 5: Evaluate the Riemann sum. Use Theorem 5.1 or a calculator to compute the value of the sum. Theorem 5.1 states that the sum of a linear function over an interval can be simplified using properties of summation. Break the sum into simpler components and compute each part.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Riemann Sums
Riemann sums are a method for approximating the definite integral of a function over an interval. They involve partitioning the interval into smaller subintervals, calculating the function's value at specific points (like the right endpoint), and summing the products of these values and the widths of the subintervals. This approach helps in understanding the area under a curve and is foundational for integral calculus.
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Sigma Notation
Sigma notation is a concise way to represent the sum of a sequence of terms. It uses the Greek letter sigma (Σ) to indicate summation, along with an index of summation that specifies the starting and ending values. In the context of Riemann sums, sigma notation allows for a clear expression of the sum of function values multiplied by subinterval widths, facilitating easier calculations and evaluations.
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Theorem 5.1
Theorem 5.1 typically refers to a specific theorem in calculus that relates Riemann sums to definite integrals, often stating that as the number of subintervals increases (n approaches infinity), the Riemann sum converges to the exact value of the integral. This theorem is crucial for understanding the fundamental connection between discrete approximations and continuous functions, reinforcing the concept of integration as the limit of Riemann sums.
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