Ch. 5 - Integrals

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 5 - Integrals

Problema 5.PE.3a

Capitolo 5, Problema 5.PE.3a

           10           10
Suppose that Σ aₖ = -2 and Σ bₖ = 25. Find the value of
           k = 1          k = 1

  10
a. Σ aₖ/4
  k = 1

Guida verificata passo dopo passo

Identify the given information: \( \sum_{k=1}^{10} a_k = -2 \) and \( \sum_{k=1}^{10} b_k = 25 \).

Understand that the problem asks for \( \sum_{k=1}^{10} \frac{a_k}{4} \), which means each term \( a_k \) is divided by 4 before summing.

Recall the property of summations that allows factoring out constants: \( \sum_{k=1}^{n} c \cdot a_k = c \cdot \sum_{k=1}^{n} a_k \), where \( c \) is a constant.

Apply this property to the given sum: \( \sum_{k=1}^{10} \frac{a_k}{4} = \frac{1}{4} \sum_{k=1}^{10} a_k \).

Substitute the known sum \( \sum_{k=1}^{10} a_k = -2 \) into the expression to write \( \sum_{k=1}^{10} \frac{a_k}{4} = \frac{1}{4} \times (-2) \).

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Summation Notation (Sigma Notation)

Summation notation uses the Greek letter sigma (Σ) to represent the sum of a sequence of terms indexed by an integer variable. It concisely expresses adding terms like a₁ + a₂ + ... + aₙ. Understanding the limits of summation and the general term is essential for manipulating and evaluating sums.

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Sigma Notation

Properties of Summations

Summations have linearity properties, meaning the sum of a constant times a sequence equals the constant times the sum of the sequence. For example, Σ (c * aₖ) = c * Σ aₖ. This property allows factoring constants out of sums, simplifying calculations involving scaled sequences.

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Properties of Functions

Given Series Sums and Their Use

Knowing the total sums of sequences (like Σ aₖ = -2) allows direct substitution when evaluating related sums. This is useful when the problem asks for sums of scaled or transformed sequences, enabling quick computation without summing individual terms.

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Intro to Series: Partial Sums

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Definite Integrals

In Exercises 5–8, express each limit as a definite integral. Then evaluate the integral to find the value of the limit. In each case, P is a partition of the given interval, and the numbers cₖ are chosen from the subintervals of P.

lim ∑ (cos(cₖ/2)) ∆xₖ, where P is a partition of [-π, 0]

∥P∥→0 k = 1

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If ∫₀² ƒ(x) dx = π, ∫₀² 7g(x) dx = 7, and ∫₀¹ g(x) dx = 2, find the value of each of the following.

b. ∫₁² g(x) dx

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In Exercises 11–14, find the total area of the region between the graph of f and the x-axis.

ƒ(x) = 5 - 5x²/³, -1 ≤ x ≤ 8

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