Textbook Question
Limits of sums Use the definition of the definite integral to evaluate the following definite integrals. Use right Riemann sums and Theorem 5.1.
β«ββ· (4π + 6) dπ
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8. Definite Integrals
Riemann Sums
3:10 minutesProblem 5.2.77b
Textbook Question
{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.
β«ββ΄ (4πβ πΒ²) dπ
Verified step by step guidance1
Step 1: Understand the problem. You are tasked with estimating the value of the definite integral β«ββ΄ (4π - πΒ²) dπ using midpoint Riemann sums with n = 20, 50, and 100. A midpoint Riemann sum approximates the area under a curve by dividing the interval into subintervals, calculating the function value at the midpoint of each subinterval, and summing the areas of the resulting rectangles.
Step 2: Divide the interval [0, 4] into n subintervals. For n = 20, 50, and 100, calculate the width of each subinterval, Ξπ, using the formula Ξπ = (b - a) / n, where [a, b] is the interval of integration. Here, a = 0 and b = 4.
Step 3: Determine the midpoints of each subinterval. For each subinterval, the midpoint is calculated as (πα΅’ββ + πα΅’) / 2, where πα΅’ββ and πα΅’ are the endpoints of the subinterval. Generate these midpoints for n = 20, 50, and 100.
Step 4: Evaluate the function f(π) = 4π - πΒ² at each midpoint. Use a calculator to compute the function values for all midpoints corresponding to n = 20, 50, and 100.
Step 5: Compute the midpoint Riemann sum for each n value. Multiply each function value by the width of the subinterval, Ξπ, and sum these products to approximate the integral. Repeat this process for n = 20, 50, and 100 to obtain increasingly accurate estimates of the integral.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integral
A definite integral represents the signed area under a curve defined by a function over a specific interval. It is denoted as β«βα΅ f(x) dx, where 'a' and 'b' are the limits of integration. The value of a definite integral can be interpreted as the accumulation of quantities, such as area, over the interval from 'a' to 'b'.
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Definition of the Definite Integral
Riemann Sum
A Riemann sum is a method for approximating the value of a definite integral by dividing the area under the curve into smaller rectangles. The sum is calculated by taking the function values at specific points (left, right, or midpoints) and multiplying by the width of the subintervals. As the number of rectangles increases (n β β), the Riemann sum approaches the exact value of the definite integral.
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Midpoint Riemann Sum
A midpoint Riemann sum 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 provides a better approximation of the integral compared to left or right endpoint evaluations, especially for functions that are continuous and smooth. The accuracy improves as the number of subintervals (n) increases.
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