Ch. 5 - Integration

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 5 - Integration

Problem 5.1.63

Chapter 5, Problem 5.1.63

{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 midpoint Riemann sum for f(x) = x³ on [3,11] with n = 32.

Verified step by step guidance

Step 1: Understand the problem. We are tasked with writing the midpoint Riemann sum for the function f(x) = x³ on the interval [3, 11] with n = 32 using sigma notation. Then, we need to evaluate the sum using Theorem 5.1 or a calculator.

Step 2: Divide the interval [3, 11] into n = 32 subintervals. The width of each subinterval, Δx, is calculated as Δx = (b - a) / n, where a = 3, b = 11, and n = 32. Use MathML:

\frac{(11 - 3)}{32}

Step 3: Determine the midpoint of each subinterval. The midpoint of the i-th subinterval is given by x_i = a + (i - 0.5)Δx, where i ranges from 1 to n. Use MathML:

x

_i=a+(i-0.5)Δx.

Step 4: Write the Riemann sum in sigma notation. The midpoint Riemann sum is expressed as S = Σ f(x_i)Δx, where i ranges from 1 to n. Substitute f(x) = x³ and Δx from Step 2 into the formula. Use MathML:

\sum i_{1} x^{3} Δx

Step 5: Evaluate the Riemann sum using Theorem 5.1 or a calculator. Plug in the values for Δx and x_i into the sigma notation and compute the sum. This step involves numerical computation, which can be done using a calculator or software.

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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 within these subintervals, and summing the products of these values and the widths of the subintervals. The midpoint Riemann sum specifically uses the midpoint of each subinterval to evaluate the function, providing a more accurate approximation than using left or right endpoints.

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. For Riemann sums, sigma notation allows us to express the sum of function values multiplied by the width of the subintervals in a compact form, making it easier to manipulate and evaluate mathematically.

Theorem 5.1 (Fundamental Theorem of Calculus)

Theorem 5.1, often referred to as the Fundamental Theorem of Calculus, establishes a connection between differentiation and integration. It states that if a function is continuous on a closed interval, then the definite integral of the function can be computed using its antiderivative. This theorem allows us to evaluate Riemann sums by finding the antiderivative of the function and applying the limits of integration, thus simplifying the process of calculating the area under the curve.

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Related Practice

Textbook Question

Variations on the substitution method Evaluate the following integrals.

∫ 𝓍/(√𝓍―4) d𝓍

Textbook Question

Multiple substitutions If necessary, use two or more substitutions to find the following integrals.

∫₀^π/² (cos θ sin θ) / √(cos² θ + 16) dθ (Hint: Begin with u = cos θ .)

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Textbook Question

Suppose an object moves along a line at 15 m/s, for 0 ≤ t < 2 and at 25 m/s, for 2 ≤ t ≤ 5, where t is measured in seconds. Sketch the graph of the velocity function and find the displacement of the object for 0 ≤ t ≤ 5.

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Textbook Question

Evaluate ∫₃⁸ ƒ ′(t) dt , where ƒ ′ is continuous on [3, 8], ƒ(3) = 4, and ƒ(8) = 20 .

Textbook Question

Definite integrals Evaluate the following integrals using the Fundamental Theorem of Calculus

∫₁⁹ 2/(√𝓍) d𝓍

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Textbook Question

When using a change of variables u = g(𝓍) to evaluate the definite integral ∫ₐᵇ ƒ(g(𝓍)) g' (𝓍) d(𝓍), how are the limits of integration transformed?

{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 midpoint Riemann sum for f(x) = x³ on [3,11] with n = 32.

Verified video answer for a similar problem:

Key Concepts

Riemann Sums

Sigma Notation

Theorem 5.1 (Fundamental Theorem of Calculus)

{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 midpoint Riemann sum for f(x) = x³ on [3,11] with n = 32.