{Use of Tech} Midpoint Riemann sums with a calculator Consider the following definite integrals.
(a) Write the midpoint Riemann sum in sigma notation for an arbitrary value of n.


∫₁⁴ 2√𝓍 d𝓍

Verified step by step guidance

1. Step 1: Understand the problem. A midpoint Riemann sum approximates the value of a definite integral by dividing the interval into n subintervals, calculating the function value at the midpoint of each subinterval, and summing the areas of the rectangles formed.
2. Step 2: Define the interval and subintervals. The integral ∫₁⁴ 2√𝓍 d𝓍 is over the interval [1, 4]. Divide this interval into n subintervals of equal width Δ𝓍 = (4 - 1)/n = 3/n.
3. Step 3: Determine the midpoints of the subintervals. The midpoint of the i-th subinterval is given by 𝓍ᵢ = 1 + (i - 0.5)Δ𝓍, where i ranges from 1 to n.
4. Step 4: Write the function value at the midpoint. The function being integrated is f(𝓍) = 2√𝓍. At the midpoint 𝓍ᵢ, the function value is f(𝓍ᵢ) = 2√(1 + (i - 0.5)Δ𝓍).
5. Step 5: Express the midpoint Riemann sum in sigma notation. The sum is approximated as Sₙ = Σ (from i=1 to n) [f(𝓍ᵢ) * Δ𝓍], which becomes Sₙ = Σ (from i=1 to n) [2√(1 + (i - 0.5)(3/n)) * (3/n)].

Key Concepts

Midpoint Riemann Sum

A Midpoint Riemann Sum is a method for approximating the value of a definite integral. It involves dividing the interval into 'n' subintervals, calculating the midpoint of each subinterval, and then evaluating the function at these midpoints. The sum of these values, multiplied by the width of the subintervals, provides an estimate of the area under the curve.

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 us to express the sum of function values at midpoints over all subintervals in a clear and compact form.

Definite Integral

A definite integral represents the signed area under a curve defined by a function over a specific interval [a, b]. It is calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. The definite integral provides a precise value that corresponds to the accumulation of quantities, such as area, over the interval, and is often evaluated using techniques like Riemann sums.