{Use of Tech} Midpoint Riemann sums with a calculator Consider the following definite integrals.
(b) Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral.


∫₁⁴ 2√𝓍 d𝓍

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to estimate the value of the definite integral ∫₁⁴ 2√𝓍 d𝓍 using midpoint Riemann sums with n = 20, 50, and 100. A midpoint Riemann sum approximates the 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: Divide the interval [1, 4] into n subintervals. The width of each subinterval, Δ𝓍, is calculated as Δ𝓍 = (4 - 1) / n. For n = 20, 50, and 100, compute Δ𝓍 for each case.
3. Step 3: Determine the midpoints of each subinterval. For each subinterval [𝓍ᵢ, 𝓍ᵢ₊₁], the midpoint is given by 𝓂ᵢ = (𝓍ᵢ + 𝓍ᵢ₊₁) / 2. Calculate the midpoints for all subintervals for n = 20, 50, and 100.
4. Step 4: Evaluate the function 2√𝓍 at each midpoint. For each midpoint 𝓂ᵢ, compute f(𝓂ᵢ) = 2√𝓂ᵢ. This gives the height of the rectangle for each subinterval.
5. Step 5: Compute the midpoint Riemann sum for each value of n. Multiply the function value at each midpoint by the width of the subinterval, Δ𝓍, and sum these products for all subintervals. This gives the approximate value of the integral for n = 20, 50, and 100.

Key Concepts

Definite Integral

A definite integral represents the signed area under a curve defined by a function over a specific interval. It is calculated using the Fundamental Theorem of Calculus, which connects differentiation and integration. The notation ∫ₐᵇ f(x) dx indicates the integral of f(x) from a to b, providing a numerical value that reflects the accumulation of quantities.

Riemann Sums

Riemann sums are a method for approximating the value of a definite integral by dividing the area under a curve into smaller rectangles. The sum of the areas of these rectangles, calculated using sample points within each subinterval, provides an estimate of the integral. The accuracy of the approximation improves as the number of rectangles (n) increases, making it essential to understand how to compute these sums effectively.

Midpoint Rule

The Midpoint Rule is a specific type of Riemann sum where the height of each rectangle is determined by the function value at the midpoint of each subinterval. This method often yields better approximations than using left or right endpoints, especially for functions that are continuous and smooth. By applying this rule with varying values of n, one can observe how the approximation converges to the actual value of the integral.