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                                                                                                                                                                                                                      

Verified step by step guidance

1. Step 1: Recognize the structure of the given limit. The expression lim ∑ 𝓍ₖ (ln 𝓍ₖ) ∆𝓍ₖ as ∆ → 0 is a Riemann sum. A Riemann sum approximates the area under a curve by summing up small rectangles, and as the width of these rectangles (∆𝓍ₖ) approaches zero, the sum converges to a definite integral.
2. Step 2: Identify the interval of integration. The problem specifies the interval [1, 2], which means the definite integral will be evaluated over this range.
3. Step 3: Determine the function ƒ(𝓍) being integrated. In the Riemann sum, the term 𝓍ₖ (ln 𝓍ₖ) corresponds to the function ƒ(𝓍) = 𝓍 ln(𝓍). This is the function that will be integrated.
4. Step 4: Write the definite integral. The limit of the Riemann sum can be expressed as the definite integral of ƒ(𝓍) = 𝓍 ln(𝓍) over the interval [1, 2]. Using proper notation, this is written as: $\int a_1{b}_{2}x\ln xdx$
5. Step 5: Conclude the process. The definite integral $\int a_1{b}_{2}x\ln xdx$ represents the limit of the given Riemann sum as ∆ → 0. This integral can now be evaluated using standard techniques of integration, if needed.

Key Concepts

Riemann Sums

Riemann sums are a method for approximating the definite integral of a function over an interval by dividing the interval into smaller subintervals. For each subinterval, a sample point is chosen, and the function's value at that point is multiplied by the width of the subinterval. As the number of subintervals increases and their width decreases, the Riemann sum approaches the exact value of the definite integral.

Definite Integrals

A definite integral represents the signed area under a curve defined by a function over a specific interval [a, b]. It is denoted as ∫[a,b] f(x) dx and can be interpreted as the limit of Riemann sums as the number of subintervals approaches infinity. Definite integrals have numerous applications in calculating areas, volumes, and solving problems in physics and engineering.

Limits

In calculus, a limit describes the behavior of a function as its input approaches a certain value. Limits are fundamental in defining both derivatives and integrals. In the context of Riemann sums, the limit is taken as the width of the subintervals approaches zero, allowing for the transition from a sum of areas of rectangles to the exact area under the curve, represented by the definite integral.