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 5.1.49d

Chapter 5, Problem 5.1.49d

Sigma notation Evaluate the following expressions.
(d)     5
       ∑ (1 + n²)
       n=1

Verified step by step guidance

Step 1: Understand the problem. The given expression is a summation in sigma notation: ∑ (1 + n²) from n = 1 to 5. This means we need to calculate the sum of the terms (1 + n²) for each integer value of n starting at 1 and ending at 5.

Step 2: Write out the individual terms of the summation. Substitute each value of n (from 1 to 5) into the expression (1 + n²). The terms will be: (1 + 1²), (1 + 2²), (1 + 3²), (1 + 4²), and (1 + 5²).

Step 3: Simplify each term. For example, (1 + 1²) simplifies to 2, (1 + 2²) simplifies to 5, and so on. Perform this simplification for all terms.

Step 4: Add the simplified terms together. Once all terms are simplified, sum them up to find the total value of the summation.

Step 5: Verify your work. Double-check each substitution, simplification, and addition to ensure accuracy in your calculations.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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, followed by an expression that defines the terms to be summed. The notation includes limits that specify the starting and ending indices of the summation, allowing for efficient calculation of series.

Series and Sequences

A sequence is an ordered list of numbers, while a series is the sum of the terms of a sequence. Understanding the difference is crucial for evaluating expressions in sigma notation. In the context of the given question, the series involves summing the values of the expression (1 + n²) for each integer n within specified limits.

Evaluating Summations

Evaluating summations involves calculating the total of a series of numbers generated by a specific formula. This requires substituting the index values into the expression and performing the arithmetic operations. Mastery of this concept is essential for accurately solving problems that involve sigma notation, such as the one presented in the question.

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Related Practice

Textbook Question

{Use of Tech} Approximating definite integrals Complete the following steps for the given integral and the given value of n.

(d) Determine which Riemann sum (left or right) underestimates the value of the definite integral and which overestimates the value of the definite integral.

∫₃⁶ (1―2𝓍) d𝓍 ; n = 6

Textbook Question

Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.

{Use of Tech} ƒ(𝓍) = e ˣ/₂ on [1,4]; n = 6

(d) Calculate the left and right Riemann sums.

Textbook Question

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

(d) If A(𝓍) = 3𝓍²― 𝓍― 3 is an area function for ƒ, then

B(𝓍) = 3𝓍² ― 𝓍 is also an area function for ƒ.

Textbook Question

Area functions The graph of ƒ is shown in the figure. Let A(x) = ∫₀ˣ ƒ(t) dt and F(x) = ∫₂ˣ ƒ(t) dt be two area functions for ƒ. Evaluate the following area functions.

(d) F(8)

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Textbook Question

Properties of integrals Suppose ∫₀³ƒ(𝓍) d𝓍 = 2 , ∫₃⁶ƒ(𝓍) d𝓍 = ―5 , and ∫₃⁶g(𝓍) d𝓍 = 1. Evaluate the following integrals.

(a) ∫₀³ 5ƒ(𝓍) d𝓍

Textbook Question

Left and right Riemann sums Complete the following steps for the given function, interval, and value of n.

{Use of Tech} ƒ(𝓍) = cos 𝓍 on [0. π/2]; n = 4

(d) Calculate the left and right Riemann sums.

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